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Unformatted text preview: 0.1 Exercises for Chapter 3 1. An unfair die with six sides is rolled. Let X be outcome of the die. The probability mass function is given to be P(X=i)=i/21 , i=1,2,....6. a) Show that this is a legitimate pmf. b) Find and plot the cumulative distribution function. 2. Put a check-mark in the table below indicating whether or not each of p 1 ( x ), p 2 ( x ), p 3 ( x ) is a legitimate probability mass function (pmf) for the random variable X . x 1 2 3 Is legitimate Is not legitimate p 1 ( x ) .2 .3 .4 .2 p 2 ( x ) .3 .3 .5 -.1 p 3 ( x ) .1 .4 .4 .1 b) Calculate E ( X ) and E (1 /X ) using the following pmf: x 1 2 3 4 p ( x ) .4 .3 .1 .2 c) In a win-win game, the player will win a monetary price, but he/she has to decide between a fixed and a random price. In particular the player is offered $1000 E ( X ) or $1000 X , where the random variable X has the distribution given in (b). When the choice is made a value of X is generated and the player receives the chosen price. Which choice would you recommend the player to make? 3. A metal fabricating plant currently has five major pieces under contract each with a deadline for completion. Let X be the number of pieces completed by their deadlines. Suppose that X is a random variable with p.m.f. p ( x ) given by x 1 2 3 4 5 p ( x ) .05 .10 .15 .25 .35 .10 (a) Find and plot the cdf of X . (b) Use the cdf to find the probability that between one and four pieces, inclusive, are completed by deadline. 1 (c) Find the expectation of X . (d) Find the variance of X . 4. Let X denote the daily sales for a computer manufacturing firm. The cumulative distribution function of the random variable X is F ( x ) = x < . 2 ≤ x < 1 . 7 1 ≤ x < 2 . 9 2 ≤ x < 3 1 3 ≤ x (a) Plot the cumulative distribution function. What is the probability of two or more sales in a day? (b) Write down the probability mass function of X ....
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This note was uploaded on 01/11/2012 for the course STAT 401 taught by Professor Akritas during the Fall '00 term at Penn State.
- Fall '00