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Assignment 1
Problem 6
(MGF of the geometric distribution) If \( X \sim \) Geometric \( (p) \), find the MGF of \( X \).
Problem 7
If \( M_{X}(s)=\frac{1}{4}+\frac{1}{2} e^{s}+\frac{1}{4} e^{2 s} \), find \( E X \) and \( \operatorname{Var}(X) \).
Problem 8
Using MGFs show that if \( X \sim N\left(\mu_{X}, \sigma_{X}^{2}\right) \) and \( Y \sim N\left(\mu_{Y}, \sigma_{Y}^{2}\right) \) are independent, then
\[
X+Y \sim N\left(\mu_{X}+\mu_{Y}, \sigma_{X}^{2}+\sigma_{Y}^{2}\right) .
\]
Problem 9
(MGF of the Laplace distribution) Let \( X \) be a continuous random variable with the following PDF
\[
f_{X}(x)=\frac{\lambda}{2} e^{-\lambda|x|}
\]
Find the MGF of \( X, M_{X}(s) \).
Problem 12
Let \( X \) be a random variable with characteristic function \( \phi_{X}(\omega) \). If \( Y=a X+b \), show that
\[
\phi_{Y}(\omega)=e^{j \omega b} \phi_{X}(a \omega)
\]
Problem 13
Let \( X \) and \( Y \) be two jointly continuous random variables with joint PDF
\[
f_{X, Y}(x, y)=\left\{\begin{array}{ll}
\frac{1}{2}(3 x+y) & 0 \leq x, y \leq 1 \\
0 & \text { otherwise }
\end{array}\right.
\]
and let the random vector \( \mathbf{U} \) be defined as
\[
U=\left[\begin{array}{l}
X \\
Y
\end{array}\right]
\]
1. Find the mean vector of \( \mathbf{U}, E \mathbf{U} \).
2. Find the correlation matrix of \( \mathbf{U}, \mathbf{R}_{\mathbf{U}} \).
3. Find the covariance matrix of \( \mathbf{U}, \mathbf{C}_{\mathbf{U}} \).
