Tutorial on Probability and Random Variables

SOATE-37

Q3. A RV has a pdf fX(x) = 2e-2x, x ≥ 0. Obtain an upper bound for P[X ≥ 1]. Obtain an upper bound for P[|X − E[X]| ≥ 1].

Q4. Given two random variables X and Y with the joint CDF FXY(x, y) and marginal CDFs FX(x) and FY(y), respectively, compute the joint probability that X is greater than a and Y is greater than b.

Q5. The joint PDF of the random variables X and Y is defined as follows:

a) Check whether X and Y are uncorrelated.

Q6. Consider four independent rolls of a 6-sided die. Let X be the number of 1’s and let Y be the number of 2’s obtained. What is the joint PMF of X and Y ?

Q7. The metro train arrives at the station near your home every quarter hour starting at 6:00 AM. You walk into the station every morning between 7:10 and 7:30 AM, with the time in this interval being a uniform random variable. What is the PDF of the time you have to wait for the first train to arrive?

Q8. Two archers shoot at a target. The distance of each shot from the center of the target is uniformly distributed from 0 to 1, independently of the other shot. What is the PDF of the distance of the losing shot from the center?

Q9. Romeo and Juliet have a date at a given time, and each, independently, will be late by an amount of time that is exponentially distributed with parameter λ. What is the PDF of the difference between their times of arrival?

Q10. Consider joint random variable (X,Y). Show that if X and Y are independent, then every event of the form (a<X≤b) is independent of the form (c<Y≤d).