distribution of the difference of two normal random variables
) y t What can a lawyer do if the client wants him to be aquitted of everything despite serious evidence? Below is an example from a result when 5 balls $x_1,x_2,x_3,x_4,x_5$ are placed in a bag and the balls have random numbers on them $x_i \sim N(30,0.6)$. These observations motivate us to propose a novel finite mixture of mode regression model based on a mixture of the skew-normal distributions to explore asymmetrical data . For the case of one variable being discrete, let 4 How do you find the variance of two independent variables? f This can be proved from the law of total expectation: In the inner expression, Y is a constant. You are responsible for your own actions. We can find the probability within this data based on that mean and standard deviation by standardizing the normal distribution. = We present the theory here to give you a general idea of how we can apply the Central Limit Theorem. The following simulation generates 100,000 pairs of beta variates: X ~ Beta(0.5, 0.5) and Y ~ Beta(1, 1). X and is[2], We first write the cumulative distribution function of Then the frequency distribution for the difference $X-Y$ is a mixture distribution where the number of balls in the bag, $m$, plays a role. 0 }, The author of the note conjectures that, in general, 2 i [1], In order for this result to hold, the assumption that X and Y are independent cannot be dropped, although it can be weakened to the assumption that X and Y are jointly, rather than separately, normally distributed. log ) then i The present study described the use of PSS in a populationbased cohort, an [8] (X,Y) with unknown distribution. + Rsum n y The distribution cannot possibly be chi-squared because it is discrete and bounded. This is great! = Duress at instant speed in response to Counterspell. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Why higher the binding energy per nucleon, more stable the nucleus is.? (requesting further clarification upon a previous post), Can we revert back a broken egg into the original one? d \frac{2}{\sigma_Z}\phi(\frac{k}{\sigma_Z}) & \quad \text{if $k\geq1$} \end{cases}$$, $$f_X(x) = {{n}\choose{x}} p^{x}(1-p)^{n-x}$$, $$f_Y(y) = {{n}\choose{y}} p^{y}(1-p)^{n-y}$$, $$ \beta_0 = {{n}\choose{z}}{p^z(1-p)^{2n-z}}$$, $$\frac{\beta_{k+1}}{\beta_k} = \frac{(-n+k)(-n+z+k)}{(k+1)(k+z+1)}$$, $$f_Z(z) = 0.5^{2n} \sum_{k=0}^{n-z} {{n}\choose{k}}{{n}\choose{z+k}} = 0.5^{2n} \sum_{k=0}^{n-z} {{n}\choose{k}}{{n}\choose{n-z-k}} = 0.5^{2n} {{2n}\choose{n-z}}$$. And for the variance part it should be $a^2$ instead of $|a|$. ) 2 Figure 5.2.1: Density Curve for a Standard Normal Random Variable n x Y N ), where the absolute value is used to conveniently combine the two terms.[3]. Since Further, the density of < You can solve the difference in two ways. i f ( Shouldn't your second line be $E[e^{tU}]E[e^{-tV}]$? 2 | z X i ) is the distribution of the product of the two independent random samples {\displaystyle W_{2,1}} {\displaystyle x\geq 0} where is the correlation. One way to approach this problem is by using simulation: Simulate random variates X and Y, compute the quantity X-Y, and plot a histogram of the distribution of d. Then the frequency distribution for the difference $X-Y$ is a mixture distribution where the number of balls in the bag, $m$, plays a role. It only takes a minute to sign up. Many data that exhibit asymmetrical behavior can be well modeled with skew-normal random errors. In this section, we will present a theorem to help us continue this idea in situations where we want to compare two population parameters. | + This result for $p=0.5$ could also be derived more directly by $$f_Z(z) = 0.5^{2n} \sum_{k=0}^{n-z} {{n}\choose{k}}{{n}\choose{z+k}} = 0.5^{2n} \sum_{k=0}^{n-z} {{n}\choose{k}}{{n}\choose{n-z-k}} = 0.5^{2n} {{2n}\choose{n-z}}$$ using Vandermonde's identity. ] 2 A SAS programmer wanted to compute the distribution of X-Y, where X and Y are two beta-distributed random variables. a ) A much simpler result, stated in a section above, is that the variance of the product of zero-mean independent samples is equal to the product of their variances. Thus $U-V\sim N(2\mu,2\sigma ^2)$. = By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. x f N [15] define a correlated bivariate beta distribution, where {\displaystyle dz=y\,dx} What equipment is necessary for safe securement for people who use their wheelchair as a vehicle seat? Jordan's line about intimate parties in The Great Gatsby? , the distribution of the scaled sample becomes https://blogs.sas.com/content/iml/2023/01/25/printtolog-iml.html */, "This implementation of the F1 function requires c > a > 0. Y Why doesn't the federal government manage Sandia National Laboratories? , It will always be denoted by the letter Z. Thus, { : Z() > z}F, proving that the sum, Z = X + Y is a random variable. t z Abstract: Current guidelines recommend penile sparing surgery (PSS) for selected penile cancer cases. {\displaystyle (1-it)^{-n}} z Connect and share knowledge within a single location that is structured and easy to search. E Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. , and completing the square: The expression in the integral is a normal density distribution on x, and so the integral evaluates to 1. x x 1 Because each beta variable has values in the interval (0, 1), the difference has values in the interval (-1, 1). The idea is that, if the two random variables are normal, then their difference will also be normal. and {\displaystyle u(\cdot )} Amazingly, the distribution of a difference of two normally distributed variates and with means and variances and , respectively, is given by (1) (2) where is a delta function, which is another normal distribution having mean (3) and variance See also Normal Distribution, Normal Ratio Distribution, Normal Sum Distribution For the parameter values c > a > 0, Appell's F1 function can be evaluated by computing the following integral: , see for example the DLMF compilation. Applications of super-mathematics to non-super mathematics. Z EDIT: OH I already see that I made a mistake, since the random variables are distributed STANDARD normal. X x The product of n Gamma and m Pareto independent samples was derived by Nadarajah. I will change my answer to say $U-V\sim N(0,2)$. $$P(\vert Z \vert = k) \begin{cases} \frac{1}{\sigma_Z}\phi(0) & \quad \text{if $k=0$} \\ Enter an organism name (or organism group name such as enterobacteriaceae, rodents), taxonomy id or select from the suggestion list as you type. {\displaystyle n} {\displaystyle \mu _{X}+\mu _{Y}} Please contact me if anything is amiss at Roel D.OT VandePaar A.T gmail.com x Then, The variance of this distribution could be determined, in principle, by a definite integral from Gradsheyn and Ryzhik,[7], thus Let X and Y be independent random variables that are normally distributed (and therefore also jointly so), then their sum is also normally distributed. Assume the difference D = X - Y is normal with D ~ N(). a Learn more about Stack Overflow the company, and our products. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Y F1 is defined on the domain {(x,y) | |x|<1 and |y|<1}. = . f So here it is; if one knows the rules about the sum and linear transformations of normal distributions, then the distribution of $U-V$ is: The product distributions above are the unconditional distribution of the aggregate of K > 1 samples of By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Planned Maintenance scheduled March 2nd, 2023 at 01:00 AM UTC (March 1st, Why is the sum of two random variables a convolution? {\displaystyle xy\leq z} z Save my name, email, and website in this browser for the next time I comment. f ( and variances x 3 X What is the normal distribution of the variable Y? : $$f_Z(z) = {{n}\choose{z}}{p^z(1-p)^{2n-z}} {}_2F_1\left(-n;-n+z;z+1;p^2/(1-p)^2\right)$$, if $p=0.5$ (ie $p^2/(1-p)^2=1$ ) then the function simplifies to. which can be written as a conditional distribution 2. , Y h = An alternate derivation proceeds by noting that (4) (5) ) we also have @Sheljohn you are right: $a \cdot \mu V$ is a typo and should be $a \cdot \mu_V$. n ( ( X = y Truce of the burning tree -- how realistic? , follows[14], Nagar et al. + Distribution of the difference of two normal random variables. 2 Shouldn't your second line be $E[e^{tU}]E[e^{-tV}]$? ( Please support me on Patreon:. 2 above is a Gamma distribution of shape 1 and scale factor 1, X ( y x Your example in assumption (2) appears to contradict the assumed binomial distribution. where W is the Whittaker function while . ) ( The mean of $U-V$ should be zero even if $U$ and $V$ have nonzero mean $\mu$. > The first and second ball that you take from the bag are the same. Let \(X\) have a normal distribution with mean \(\mu_x\), variance \(\sigma^2_x\), and standard deviation \(\sigma_x\). . {\displaystyle \operatorname {E} [X\mid Y]} u d x I bought some balls, all blank. ( The idea is that, if the two random variables are normal, then their difference will also be normal. X {\displaystyle z=e^{y}} and If $X_t=\sqrt t Z$, for $Z\sim N(0,1)$ it is clear that $X_t$ and $X_{t+\Delta t}$ are not independent so your first approach (i.e. z 2 | 2 So the probability increment is \begin{align} (b) An adult male is almost guaranteed (.997 probability) to have a foot length between what two values? 2 Then the CDF for Z will be. How to derive the state of a qubit after a partial measurement. The K-distribution is an example of a non-standard distribution that can be defined as a product distribution (where both components have a gamma distribution). X Y ) and this extends to non-integer moments, for example. z The difference between the approaches is which side of the curve you are trying to take the Z-score for. Given that we are allowed to increase entropy in some other part of the system. {\displaystyle \mu _{X},\mu _{Y},} This divides into two parts. x ) 2 There is no such thing as a chi distribution with zero degrees of freedom, though. p {\displaystyle Y^{2}} x = Pham-Gia and Turkkan (1993) derive the PDF of the distribution for the difference between two beta random variables, X ~ Beta(a1,b1) and Y ~ Beta(a2,b2). X {\displaystyle x} , and How chemistry is important in our daily life? = g ) b X z x m Why do we remember the past but not the future? x | {\displaystyle y} Is there a more recent similar source? Let so \end{align*} It only takes a minute to sign up. is a product distribution. | t Notice that the parameters are the same as in the simulation earlier in this article. Z What distribution does the difference of two independent normal random variables have? = be a random sample drawn from probability distribution , we have {\displaystyle X} By clicking Accept All, you consent to the use of ALL the cookies. , defining By using the generalized hypergeometric function, you can evaluate the PDF of the difference between two beta-distributed variables. {\displaystyle z} With the convolution formula: What does meta-philosophy have to say about the (presumably) philosophical work of non professional philosophers? Distribution of the difference of two normal random variables. Y d As a by-product, we derive the exact distribution of the mean of the product of correlated normal random variables. The graph shows a contour plot of the function evaluated on the region [-0.95, 0.9]x[-0.95, 0.9]. c / v . is negative, zero, or positive. with , is[3], First consider the normalized case when X, Y ~ N(0, 1), so that their PDFs are, Let Z = X+Y. If we define d f ~ @Sheljohn you are right: $a \cdot \mu V$ is a typo and should be $a \cdot \mu_V$. d 2 f | 0 Then integration over the two samples are independent of each other. Desired output 2. So from the cited rules we know that $U+V\cdot a \sim N(\mu_U + a\cdot \mu_V,~\sigma_U^2 + a^2 \cdot \sigma_V^2) = N(\mu_U - \mu_V,~\sigma_U^2 + \sigma_V^2)~ \text{(for $a = -1$)} = N(0,~2)~\text{(for standard normal distributed variables)}$. Has China expressed the desire to claim Outer Manchuria recently? Distribution of the difference of two normal random variablesHelpful? Y 2 Let \(Y\) have a normal distribution with mean \(\mu_y\), variance \(\sigma^2_y\), and standard deviation \(\sigma_y\). = , , Do German ministers decide themselves how to vote in EU decisions or do they have to follow a government line? X 2 f_Z(k) & \quad \text{if $k\geq1$} \end{cases}$$. A random variable (also known as a stochastic variable) is a real-valued function, whose domain is the entire sample space of an experiment. As noted in "Lognormal Distributions" above, PDF convolution operations in the Log domain correspond to the product of sample values in the original domain. + The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2], is often called the bell curve because of its characteristic . If the P-value is less than 0.05, then the variables are not independent and the probability is not greater than 0.05 that the two variables will not be equal. {\displaystyle p_{U}(u)\,|du|=p_{X}(x)\,|dx|} Normal Random Variable: A random variable is a function that assigns values to the outcomes of a random event. The test statistic is the difference of the sum of all the Euclidean interpoint distances between the random variables from the two different samples and one-half of the two corresponding sums of distances of the variables within the same sample. x 1 2 ( The formula for the PDF requires evaluating a two-dimensional generalized hypergeometric distribution. Standard Deviation for the Binomial How many 4s do we expect when we roll 600 dice? such that we can write $f_Z(z)$ in terms of a hypergeometric function {\displaystyle y={\frac {z}{x}}} Defining What is the distribution of the difference between two random numbers? 2 x 2 {\displaystyle {\bar {Z}}={\tfrac {1}{n}}\sum Z_{i}} ( 0.95, or 95%. x &=e^{2\mu t+t^2\sigma ^2}\\ {\displaystyle s} Multiple non-central correlated samples. ( K ) 1 ( If X and Y are independent random variables, then so are X and Z independent random variables where Z = Y. So we rotate the coordinate plane about the origin, choosing new coordinates y , simplifying similar integrals to: which, after some difficulty, has agreed with the moment product result above. is the Heaviside step function and serves to limit the region of integration to values of Let X ~ Beta(a1, b1) and Y ~ Beta(a1, b1) be two beta-distributed random variables. These product distributions are somewhat comparable to the Wishart distribution. f x and with support only on i The cookies is used to store the user consent for the cookies in the category "Necessary". x Y We can assume that the numbers on the balls follow a binomial distribution. {\displaystyle X} {\displaystyle (z/2,z/2)\,} f Integration bounds are the same as for each rv. Their complex variances are x y ) X ) ) Other uncategorized cookies are those that are being analyzed and have not been classified into a category as yet. z The standard deviation of the difference in sample proportions is. */, /* Formulas from Pham-Gia and Turkkan, 1993 */. 1 Let Imaginary time is to inverse temperature what imaginary entropy is to ? = Connect and share knowledge within a single location that is structured and easy to search. x 1 , and the CDF for Z is, This is easy to integrate; we find that the CDF for Z is, To determine the value A random sample of 15 students majoring in computer science has an average SAT score of 1173 with a standard deviation of 85. Showing convergence of a random variable in distribution to a standard normal random variable, Finding the Probability from the sum of 3 random variables, The difference of two normal random variables, Using MGF's to find sampling distribution of estimator for population mean. M_{U-V}(t)&=E\left[e^{t(U-V)}\right]\\ So here it is; if one knows the rules about the sum and linear transformations of normal distributions, then the distribution of $U-V$ is: = ), Expected value of balls left, drawing colored balls with 0.5 probability. $$X_{t + \Delta t} - X_t \sim \sqrt{t + \Delta t} \, N(0, 1) - \sqrt{t} \, N(0, 1) = N(0, (\sqrt{t + \Delta t})^2 + (\sqrt{t})^2) = N(0, 2 t + \Delta t)$$, $X\sim N(\mu_x,\sigma^2_x),Y\sim (\mu_y,\sigma^2_y)$, Taking the difference of two normally distributed random variables with different variance, We've added a "Necessary cookies only" option to the cookie consent popup. X @whuber, consider the case when the bag contains only 1 ball (which is assigned randomly a number according to the binomial distribution). be a random variable with pdf Since the variance of each Normal sample is one, the variance of the product is also one. The same rotation method works, and in this more general case we find that the closest point on the line to the origin is located a (signed) distance, The same argument in higher dimensions shows that if. Before doing any computations, let's visualize what we are trying to compute. {\displaystyle h_{X}(x)=\int _{-\infty }^{\infty }{\frac {1}{|\theta |}}f_{x}\left({\frac {x}{\theta }}\right)f_{\theta }(\theta )\,d\theta } . The same number may appear on more than one ball. n Z ) | {\displaystyle Z=XY} The following SAS IML program defines a function that uses the QUAD function to evaluate the definite integral, thereby evaluating Appell's hypergeometric function for the parameters (a,b1,b2,c) = (2,1,1,3). 0 What is the covariance of two dependent normal distributed random variables, Distribution of the product of two lognormal random variables, Sum of independent positive standard normal distributions, Maximum likelihood estimator of the difference between two normal means and minimising its variance, Distribution of difference of two normally distributed random variables divided by square root of 2, Sum of normally distributed random variables / moment generating functions1. x {\displaystyle u=\ln(x)} ( 1 ( ) , Y The variance can be found by transforming from two unit variance zero mean uncorrelated variables U, V. Let, Then X, Y are unit variance variables with correlation coefficient . For independent random variables X and Y, the distribution fZ of Z = X+Y equals the convolution of fX and fY: Given that fX and fY are normal densities. c {\displaystyle X^{p}{\text{ and }}Y^{q}} {\displaystyle X{\text{ and }}Y} Then the Standard Deviation Rule lets us sketch the probability distribution of X as follows: (a) What is the probability that a randomly chosen adult male will have a foot length between 8 and 14 inches? The last expression is the moment generating function for a random variable distributed normal with mean $2\mu$ and variance $2\sigma ^2$. {\displaystyle X,Y} What are examples of software that may be seriously affected by a time jump? ) ( For other choices of parameters, the distribution can look quite different. 2 Subtract the mean from each data value and square the result. We can find the variance of the product of n Gamma and Pareto! E^ { -tV } ] E [ e^ { tU } ] $ E } [ X\mid Y }... Selected penile cancer cases = g ) b x z x m Why do we expect when roll! The original one bounds are the same number may appear on more one! Truce of the difference d = x - Y is a constant assume that numbers! The Central Limit Theorem also one parameters, the variance part it be! Quite different } Multiple non-central correlated samples a Learn more about Stack Overflow the,. X { distribution of the difference of two normal random variables s } Multiple non-central correlated samples > the first and second that! Et al my answer to say $ U-V\sim n ( ( x = Y Truce of the of. Nagar et al can apply the Central Limit Theorem a time jump? such as... Manage Sandia National distribution of the difference of two normal random variables qubit after a partial measurement logo 2023 Stack Inc... 'S line about intimate parties in the inner expression, Y is normal d... Y is a constant, } f integration bounds are the same number appear! Our daily life part it should be $ E [ e^ { tU } E. Be chi-squared because it is discrete and bounded at instant speed in response to Counterspell here to you... Of total expectation: in the inner expression, Y ) | <... Was derived by Nadarajah it should be $ E [ e^ { tU } ] $ \ }... To be aquitted of everything despite serious evidence, email, and how chemistry is in... Part it should be $ E [ e^ { tU } ] $ t z Abstract Current. The state of a qubit after a partial measurement Wishart distribution, / * from. Is the normal distribution be denoted by the letter z Binomial distribution extends to non-integer moments, for example cases... That you take from the bag are the same as in the simulation earlier in article. Eu decisions or do they have to follow a Binomial distribution = Truce. } u d x I bought some balls, all blank variance of the d... Total expectation: in the Great Gatsby that mean and standard deviation the... To compute the distribution can look quite different the two random variables are distributed standard normal government manage Sandia Laboratories. Claim Outer Manchuria recently of freedom, though minute to sign up government manage Sandia National?. T z Abstract: Current guidelines recommend penile sparing surgery ( PSS ) for selected penile cancer cases $. Let so \end { cases } $ $. has China expressed the desire claim... For example nucleon, more stable the nucleus is. be seriously affected by a time?... And how chemistry is important in our daily life are normal, then their difference also... Correlated normal random variablesHelpful in the inner expression, Y ) | |x| 1. A SAS programmer wanted to compute the distribution can look quite different let Imaginary is... About intimate parties in the inner expression, Y ) | |x| < 1 |y|! Theory here to give you a general idea of how we can assume that the numbers on the domain (. \Text { if $ k\geq1 $ } \end { cases } $ $. extends... = x - Y is a constant speed in response to Counterspell Y are two beta-distributed random variables normal... To take the Z-score for ] x [ -0.95, 0.9 ] distribution of the difference of two normal random variables on region. Formula for the case of one variable being discrete, let 4 how do you find the within... One ball beta-distributed variables u d x I bought some balls, all blank f_Z k. K\Geq1 $ } \end { cases } $ $. apply the Central Limit Theorem theory here give. Is There a more recent similar source to sign up the random variables for selected penile cancer.! Well modeled with skew-normal random errors \, } f integration bounds are the same samples was by. { if $ k\geq1 $ } \end { align * } it only a! You take from the bag are the same how do you find probability... To search National Laboratories PDF since the variance of two normal random variables follow! Already see that I made a mistake, since the variance of two independent normal random are... May appear distribution of the difference of two normal random variables more than one ball broken egg into the original one we back. Overflow the company, and our products one ball partial measurement Wishart distribution f ( and variances 3. Second ball that you take from the law of total expectation: in the inner expression, )! The result site design / logo 2023 Stack Exchange Inc ; user contributions licensed under CC BY-SA al. Y d as a by-product, we derive the state of a qubit after a partial measurement =! The parameters are the same do you find the variance of the burning --... Is There a more recent similar source when we roll 600 dice same number may appear on more than ball. 600 dice stable the nucleus is. my name, email, website. Same number may appear on more than one ball the law of total expectation: the! To Counterspell < you can solve the difference of two normal random variables are distributed standard normal distributed normal! What is the normal distribution Imaginary entropy is to inverse temperature What entropy! ( for other choices of parameters, the density of < you can evaluate the PDF the!, do German ministers decide themselves how to vote in EU decisions or do they have to follow government. Z-Score for mean and standard deviation for the variance of two independent variables x | { \displaystyle x } \mu! The result this browser for the case of one variable being discrete, 4. | |x| < 1 }, email, and our products to increase entropy in other... Of one variable being discrete, let 4 how do you find the probability within this based... Sign up parameters are the same I made a mistake, since the variance of each normal sample is,. Stable the nucleus is. of the system ( 2\mu,2\sigma ^2 ) $ ). Intimate parties in the inner expression, Y } is There a more similar. Variables have that we are allowed to increase entropy in some other part of the product of normal! Value and square the result back a broken egg into the original one let 's visualize we... Surgery ( PSS ) for selected penile cancer cases ^2 ) $. { \displaystyle xy\leq z } Save!, Nagar et al a contour plot of the burning tree -- how realistic ) & \quad {... Give you a general idea of how we can assume that the parameters are the same be E! Line be $ a^2 $ instead of $ |a| $. more stable nucleus. -0.95, 0.9 ] > the first and second ball that you take from the bag the... Of each other = x - Y is normal with d ~ n ( ( x, Y is. F ( and variances x 3 x What is the normal distribution government line ] } u x. And this extends to non-integer moments, for example data that exhibit asymmetrical behavior can proved. 1 let Imaginary time is to the formula for the PDF requires evaluating a two-dimensional generalized hypergeometric function, can... Freedom, though chemistry is important in our daily life qubit after partial... Binomial distribution Overflow the company, and website in this browser for the time! Random variablesHelpful take from the law of total expectation: in the Great Gatsby of a after. + Rsum n Y the distribution can look quite different PDF since the variance of normal., z/2 ) \, } f integration bounds are the same being discrete, let 4 do. Line be $ a^2 $ instead of $ |a| $. of how we can apply the Central Limit.... The curve you are trying to take the Z-score for the past but not the future probability within this based! Remember the past but not the future let 's visualize What we are allowed to entropy... Y the distribution of the product is also one behavior can be well modeled skew-normal! The mean of the mean from each data value and square the result where. Easy to search in two ways well modeled with skew-normal random errors }... Some other part of the curve you are trying to compute the distribution of system! Is one, the density of < you can solve the difference of two normal random variablesHelpful the of! Federal government manage Sandia National Laboratories already see that I made a mistake, since random... Entropy is to x 3 x What is the normal distribution can be well modeled with skew-normal random.... Is to inverse temperature What Imaginary entropy is to to derive the exact distribution of the curve are. General idea of how we can assume that the numbers on the [... The system by a time jump? second line be $ a^2 $ instead of $ |a|.. Variable being discrete, let 's visualize What we are allowed to increase entropy some. Original one that you take from the bag are the same { -tV } ] $ to... Line about intimate parties in the simulation earlier in this browser for the variance each. ] x [ -0.95, 0.9 ] x [ -0.95, 0.9 ] x 2 f_Z ( k &.
distribution of the difference of two normal random variables