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46 12:54 Edit 39 X 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}} $.

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12:54
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39
X
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 \( \mathbf{U}, \mathbf{C}_{\mathbf{U}} \).

$46 12:54 Edit 39 X Assignment 1 Problem 6 (MGF of the geometric distribution) If X ∼ Geometric (p), find the MGF of X. Problem 7 If MX(s)=(1)/(4)+(1)/(2) e^s+(1)/(4) e^2 s, find E X and Var(X). Problem 8 Using MGFs show that if X ∼ N(μX, σX^2) and Y ∼ N(μY, σY^2) are independent, then X+Y ∼ N(μX+μY, σX^2+σY^2) . Problem 9 (MGF of the Laplace distribution) Let X be a continuous random variable with the following PDF fX(x)=(λ)/(2) e^-λ|x| Find the MGF of X, MX(s). Problem 12 Let X be a random variable with characteristic function ϕX(ω). If Y=a X+b, show that ϕY(ω)=e^j ω bϕX(a ω) Problem 13 Let X and Y be two jointly continuous random variables with joint PDF fX, Y(x, y)={ (1)/(2)(3 x+y) 0 ≤ x, y ≤ 1 0 otherwise . and let the random vector 𝐔 be defined as U=[ X Y ] 1. Find the mean vector of 𝐔, E 𝐔. 2. Find the correlation matrix of 𝐔, 𝐑𝐔. 3. Find the covariance matrix of 𝐔, 𝐂𝐔.$

Added by Siphuthando S.

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