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2. Let X be Gaussian with mean $\mu_x$ and variance $\sigma_x^2$, and let Y be Gaussian with mean $\mu_y$ and variance $\sigma_y^2$. Show that if X, Y are assumed independent, then Z = X + Y is Gaussian with mean $\mu_x + \mu_y$ and variance $\sigma_x^2 + \sigma_y^2$. Hint: Write X = \bar{X} + $\mu_x$, Y = \bar{Y} + $\mu_y$ and show, using the convolution identity, that the law of Z = \bar{X} + \bar{Y} is Gaussian with mean 0 and variance $\sigma_x^2 + \sigma_y^2$.

          2. Let X be Gaussian with mean $\mu_x$ and variance $\sigma_x^2$, and let Y be Gaussian with mean $\mu_y$ and variance $\sigma_y^2$. Show that if X, Y are assumed independent, then Z = X + Y is Gaussian with mean $\mu_x + \mu_y$ and variance $\sigma_x^2 + \sigma_y^2$.
Hint: Write X = \bar{X} + $\mu_x$, Y = \bar{Y} + $\mu_y$ and show, using the convolution identity, that the law of Z = \bar{X} + \bar{Y} is Gaussian with mean 0 and variance $\sigma_x^2 + \sigma_y^2$.

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2. Let X be Gaussian with mean and variance ^2, and let Y be Gaussian with mean and variance ^2. Show that if X, Y are assumed independent, then Z = X + Y is Gaussian with mean + and variance ^2 + ^2.
Hint: Write X = X̅ + , Y = Y̅ + and show, using the convolution identity, that the law of Z = X̅ + Y̅ is Gaussian with mean 0 and variance ^2 + ^2.

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Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Sri K

01/04/2023

Step 1

Step 1: Write x = \bar{x} + \mu_(x) and Y = \bar{Y} + \mu_(Y), where \bar{x} and \bar{Y} are zero-mean Gaussian random variables. Show more…

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Let x be Gaussian with mean mu _(x) and variance sigma _(x)^(2), and let Y be Gaussian with mean mu _(Y) and variance sigma _(Y)^(2). Show that if x,Y are assumed independent, then Z=x+Y is Gaussian with mean mu _(x)+mu _(Y) and variance sigma _(x)^(2)+sigma _(Y)^(2). Hint: Write x=ar{x} +mu _(x),Y()/(b)=ar (Y)+mu _(Y) and show, using the convolution identity, that the law of /bar (Z)=ar{x} (+)/(b)ar (Y) is Gaussian with mean 0 and variance sigma _(x)^(2)+sigma _(Y)^(2). 2. Let X be Gaussian with mean x and variance ox, and let Y be Gaussian with mean y and variance y. Show that if X,Y are assumed independent, then Z = X + Y is Gaussian with mean x + Y and variance ox +oy. Hint: Write X = X + x,Y = Y + y and show, using the convolution identity, that the law of Z = X + Y is Gaussian with mean 0 and variance ox + oy.

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