Assignment Homework 1:
(1) Let f(x) = a(e-2x e-3x) for x 0 and 0 elsewhere. Find the constant a so that f(x) satisfies the
conditions of being a probability density function. Calculate Pr(X 1). (Ans a = 6; Pr(X 1) = 0.6936) Find the distribution function of X and calculate Pr ( X ). (Ans F(x) = 0 for x < 0, .75x2 + .25x for 0 x 1 and 1 for x > 1. Pr( X ) = 0.5)
(2) Let f(x) = 1.5x + .25, 0 x 1, zero elsewhere, be the probability density function of X.
(3) You play a slot machine repeatedly. The probability of winning on a single play is 0.05, and successive plays are independent. Let X be the random variable for the number of unsuccessful attempts before the first win. Find F(x) for x = 0,1,2, , and determine how many times you would need to play the slot machine in order to be sure that your probability of winning at least once is 99%. (Ans. F(x) = 1 - .95x+1, x = 0,1,2, . You need 90 plays for the probability to be 99% that you win at least once.)
(4) Consider the following experiment: Start by tossing a fair coin. If it comes up heads, toss it
again and take the result of the experiment be to a 1 if you get heads and a 0 if tails. But if the original toss comes up tails, then chose a number from the uniform distribution on (0,1) and let that be the result of the experiment. Find the distribution function of the outcome of this experiment and sketch its graph. [Ans F(x) = 0 if x < 0, F(x) = if x = 0, F(x) = (2x+1)/4 if 0<x<1, F(x) = 1 if x 1] Work (5a) if you have taken STOR 372; otherwise work (5b). (5a) Read through the solution to Example 4.2.4 on pages 103-105 in the textbook. Find the probability density function of Z. Are there any points at which f(z) is not defined? Use f(z) to find E(Z). You should get the same answer that is derived in part (a) of the solution. Ans: f(z) = 0 .1813 .4z-.6 for z = 0 for 0 < z < v5 for z < 0 and z > v5 it is undefined at z = v5
(5b) The distribution function of X is given by 1 1/x F(x) = 0 for x 1 for x < 1
Find the probability density function of X and sketch the graphs of both F(x) and f(x) . Are there any points at which f(x) is not defined? Ans: f(x) = 0 for x < 1 1/x2 for x > 1 it is undefined at x = 1