Exercise 3.1: Derive the moment generating function of the multivariate normal distribution $N(\mu, \Sigma)$.
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Step 1: The moment generating function (MGF) of a random variable X is defined as: $$M_X(t) = E[e^{tX}]$$ where E denotes the expected value. Show moreā¦
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The joint moment generating function of two random variables $X$ and $Y$ is defined to be the function $M(s, t)$ of two real variables defined by $$ M(s, t)=\mathbb{E}\left(e^{s X+t Y}\right) $$ for all values of $s$ and $t$ for which this expectation exists. Show that the joint moment generating function of a pair of random variables having the standard bivariate normal distribution (6.73) is $$ M(s, t)=\exp \left[\frac{1}{2}\left(s^{2}+2 \rho s t+t^{2}\right)\right] $$ Deduce the joint moment generating function of a pair of random variables having the bivariate normal distribution (6.76) with parameters $\mu_{1}, \mu_{2}, \sigma_{1}, \sigma_{2}, \rho$.
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Yi Z.
Suppose $X_1$ and $X_2$ are independent random variables. (i) Prove that the following relationship between the joint and marginal moment generating functions holds: $M_Y(t) = M_{X_1}(t) cdot M_{X_2}(t)$ where $Y = X_1 + X_2$ (ii) Suppose now that the distribution of $X_i$ is $N(mu_i, sigma_i^2)$, $i = 1,2$. Using the result in part (i), find the distribution of $Y = a_1X_1 + a_2X_2$ (iii) Suppose now that $X_1 sim N(3,9)$ and $X_2 sim N(2,16)$. Find the probability $Pr(X_1 - X_2 < 2)$
Sri K.
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