Course Name: Probability and Random Variables

Course Code: $110713120$

Instructor: Dr$.$ Naser Hamad and Dr$.$ Sanaa Salama

Assignment Deadline:

The assignment should be submitted by ${21}^{st}$ of May.. $2024.$ No assignments are accepted after the deadline.

Assignment Groups:

Each student is required to carry out the assignment by himself.

Expected Outcomes:

It is assumed that each student should submit the following:

a$)$ Well$-$prepared document, solving all the assigned problems.

Notes:

Upload your answer sheet on E$-$Moodle, save file as:

$110713120\_$Assignment $(1).$

The maximum points total in this assignment is $(10$ points$).$

Directions:

Read and print your name and write your signature below as indicated.

I $(.......................................................$ received a copy of assignment $(1)$ for $110713120.$

I understand that I may not copy or distribute this work to anyone.

I understand that I must return the assignment to my instructor, by the stated deadline.

I understand this is an assignment and I cannot plagiarize my answers and I cannot copy answers from other students.

I understand that I may not discuss this work with other students $($current or previous$).$

I understand that violation of this agreement will result in an $F$ on this assignment and it cannot be replaced, it will be averaged in $($as a $0\%).$

Print your name here:

Signature:

$8.$ Problems:

Problem $1$:

A random variable $x$ whose probability density function is given by

$f_x(x)=0\mathrm{.}3(x-2)+$ktri$(\frac{x-2}{3})$

Another random variable $Y$ is related to $x$ through $Y=2x-3.$ Find

The value of $k$

The expected value and the variance of $x$

The probability density function of $Y.$

Problem $2$:

DRV $x$ and $Y$ have PDFs as

$f_x(x)=0\mathrm{.}2(x)+0\mathrm{.}5(x-4)+0\mathrm{.}3(x-6)$

$f_Y(y)=0\mathrm{.}5(x+1)+0\mathrm{.}5(x-1)$

What are the expected values $E[x]$ and $E[Y]?$

What are the variances ${}_x^2$ and ${}_Y^2?$

Assume a random variable $Z=x+Y,$ what is the PDF of the random variable $Z.$