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26M1/STA21M1
Assignment Marks 50 Due date: 31 March 2023
1. Let \( \mathrm{X} \) be the sum of the up faces on a roll of a pair of fair 6 -sided dice.
a. Determine the \( p m f \) of \( X \). b. Suppose \( A=\{x: x=7,11\} \) and \( B=\{x: x=2,3,12\} \). Then, using the values of \( f(x) \) in (a), calculate \( P(A) \) and \( \mathrm{P}(\mathrm{B}) \).
c. Use also (a) to determine cdf of \( \mathrm{X} \).
\( (4) \)
2. 2. Let \( f(x)=x / 15, x=1,2,3,4,5 \), zero elsewhere, be the pmf of \( X \). Find
a. \( \mathrm{P}(\mathrm{X}=1 \) or 2\( ) \)
(3)
b. \( \mathrm{P}(1 / 2<\mathrm{X}<5 / 2) \)
(3)
\( \begin{array}{ll}\text { C. } & \mathrm{P}(1 \leq \mathrm{X} \leq 2) \text {. } \\ \text { 3. The density function for the random variable } X \text { is given by }\end{array} \)
(3)
\( f(x)=\left\{\begin{array}{ll}k x & \text { for } \quad 0 \leq x \leq 2 \\ 0 & \text { otherwise }\end{array}\right. \)
a. Find \( k \) and \( P\left(\frac{1}{2} \leq x \leq 1\right) \)
b. Find \( F_{X}(x) \)
4. A continuous random variable \( \mathrm{X} \) has pdf \( f(x)=\left\{\begin{array}{ll}k x, & 0 \leq x \leq 2 \\ k(4-x), \quad 2 \leq x \leq 4 \\ 0, & \text { otherwise }\end{array}\right. \)
a. Find the value of \( k \)
b. Find \( P\left(\frac{1}{2} \leq x \leq 2 \frac{1}{2}\right) \)
5. Let \( \mathrm{X} \) be a continuous random variable with the pdf \( f(x)=2 x \) which has support on the interval
\( (0,1) \). Suppose \( Y=\frac{1}{1+X} \), Compute \( E(Y) \).
6. Find the moment generating function of the discrete random variable \( \mathrm{X} \) which has probability distribution \( f(x)=2\left(\frac{1}{3}\right)^{x}, \quad x=1,2,3, \ldots \)
and use it to determine \( \mu_{1}^{\prime} \) and \( \mu_{2}^{\prime} \).
\( (5) \)
7. Find the moment generating function of the continuous random variable \( \mathrm{X} \) whose probability density is given by \( f(x)=\left\{\begin{array}{ll}1 & \text { for } 0<x<1 \\ 0 & \text { elsewhere }\end{array}\right. \)
and use it to determine the values of \( \mu_{1}^{\prime}, \mu_{2}^{\prime} \) and \( \sigma^{2} \).
(5)
8. Show that if a random variable has the probability density function \( f(x)=\frac{1}{2} e^{-|x|} \quad \) for \( -\infty<x<\infty \) its moment generating function is given by \( M_{X}(t)=\frac{1}{1-t^{2}} \)
