Assignment 1

Assigned 2024-01-17. Due 2024-01-24 at noon on Gradescope

Question 1

Find the characteristic function and moment generating function of the following random variables:

  1. A random variable that is uniformly distributed on the interval $[a, b]$.

  2. A random variable that is exponentially distributed with parameter $\lambda$.

Question 2

Let $X$ be a random variable, $\alpha \in \R$ and set $Y = \alpha X$. Find $M_Y$ and $\varphi_Y$ in terms of $M_X$ and $\varphi_X$.

Question 3

Let $X, Y$ be two independent random variables. Show $\varphi_{X + Y}(\lambda) = \varphi_X(\lambda) \varphi_Y(\lambda)$.

Question 4

If $X, Y$ are two random variables. The joint characteristic function of $X, Y$ is defined by $\varphi_{X, Y}(\lambda, \mu) = \E e^{i(\lambda X + \mu Y)}$. If $X, Y$ are independent, show that $\varphi_{X, Y}(\lambda, \mu) = \varphi_X(\lambda) \varphi_Y(\mu)$.

[The converse is also true: Namely, if $\varphi_{X, Y}(\lambda, \mu) = \varphi_X(\lambda) \varphi_Y(\mu)$ for every $\lambda, \mu \in \R$, then $X$ and $Y$ must be independent. This however is not as easy to prove.]

Question 5

Let $X \sim \mathcal N(0,1)$, and $Z$ be an independent random variable with $\P(Z = 1) = \P(Z = -1) = 1/2$. Let $Y = XZ$.

  1. Find the distribution of $Y$. [Hint: Compute the characteristic function.]

  2. Compute $\varphi_{X, Y}(\lambda, \mu)$ and $\varphi_X(\lambda) \varphi_Y(\mu)$.

  3. Does $\varphi_{X + Y}(\lambda) = \varphi_X(\lambda) \varphi_Y(\lambda)$? Is this consistent with Question 3?