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Problem 1

(a) Let A and B be events, where B ? A. Show that Pr[AB ? ] = Pr[A] ? Pr[B].

(b) Show that event S is independent of any event M ? S.

(c) Suppose you toss coins three times. Let X be the event that first of the three coin tosses results in "heads." Let Y be the event that you observe total of 2 "heads." Are X and Y independent events? Justify your answer.

Problem 2

Let X be a random variable uniformly distributed between -3 and 2.
(a) Define a new random variable Y = |X|. Find fY (y). (reminder: specify the ranges of Y )

(b) Find the mean of Y .

(c) Let M be the event that Y > 1. Find fX (x|M ). (reminder: specify the ranges of Y )

 

Problem 3

There is one traffic light between Kettering Laboratory and your dorm room. The green light lasts 45 seconds, yellow light lasts 15 seconds, and the red light lasts 3 minutes. If the traffic light is green or yellow, you can drive home in 5 minutes. If the traffic light is red, you must wait until it turns green.

(a) What is the probability that you return home in less than 7 minutes?
(hint 1: what is the probability that the traffic light is red?)
(hint 2: what is the distribution of the wait time if stopped at a red light?)

(b) Let X be the random variable representing the time it takes to drive home. Find the prob- ability density function of X.

(c) What is the expected driving time?

Electrical & Electronics, Engineering

  • Category:- Electrical & Electronics
  • Reference No.:- M9700857

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