# Homework 4 _SOLUTION

Notes: “o.w.” is my abbreviation for “otherwise”. This material is covered in Walpole Sections 3.1, 3.2, and 3.3.1. (From Tolga Tasdizen) (a) The function f(x) = 1.5 − 2|x|, −1 ≤ x ≤ 1 0, o.w. is not a valid probability density function. Why not? (b) The function F(x) = 1+sin(x) is not a valid cumulative distribution function. Why not? (c) The function F(x) = 0, x < 1 2 − 1 x , x ≥ 1 is not a valid cumulative distribution function. Why not? (d) Is the following valid probability mass function for a discrete random variable? If it is not, state the reason. f(x) =       0.2, x = 0 0.6, x = 1 0.2, x = 2 0, o.w. 2. Walpole Exercise 3.6. Copied here: The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the density function f(x) =( 20,000 (x+100)3 , x 0 0, o.w. Find the probability that a bottle of this medicine will have a shell life of (a) at least 200 days; and (b) anywhere from 80 to 120 days. 3. Consider the following probability mass function for a discrete random variable: f(x) =       2k, x = −1 0.5, x = 0 3k, x = 1 0, o.w. Find the value k which makes f(x) a valid probability mass function. 4. (From Tolga Tasdizen) A bag contains 3 red and 3 blue marbles. You draw three marbles from this bag without replacement. Let X be the number of blue marbles you get. (a) Is X a discrete or a continuous random variable? ECE 3530 / CS 3130 Spring 2015 2 (b) Determine the probability mass function f(x) and the cumulative distribution function F(x) for the random variable X. (c) Using F(x) ﬁnd P [X ≤ 1]. 5. Let the CDF of the lifetime in years of a microwave oven, which we denote random variable X, be: F(x) = 1 − exp(−x/4), x 0 0, o.w. (a) Is X a discrete or continuous random variable? (b) Find P [X 5]. (c) Find the probability density function of X.