Midterm%201%20Solutions

Midterm%201%20Solutions - Midterm 1 Solutions 1 Suppose...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Midterm 1 Solutions 1. Suppose that the time X (in hours) between successive commercials on a network television station has the probability density function f X ( x ) = 5 4 - x 3 , where 0 < x < 1 . (a) Find the cumulative distribution function of X . Solution. F X ( x ) = R x 0 f X ( t ) dt = R x 0 ( 5 4 - t 3 ) dt = ( 5 4 t - 1 4 t 4 ) x 0 = 5 4 x - 1 4 x 4 . (b) On average, how many hours of television uninterrupted by commercials would a viewer of this station enjoy? Solution. μ X = R 1 0 xf X ( x ) dx = R 1 0 x · ( 5 4 - x 3 ) dx = ( 5 8 x 2 - 1 5 x 5 ) 1 0 = 17 40 = 0 . 425 . (c) What is the variance of the time in minutes between successive commercials on this station? Solution. Var( X ) = R 1 0 x 2 f X ( x ) dx - μ 2 X = R 1 0 ( 5 4 x 2 - x 5 ) dx - ( 17 40 ) 2 = ( 5 12 x 3 - 1 6 x 6 ) 1 0 - ( 17 40 ) 2 = 1 4 - ( 17 40 ) 2 0 . 069 . If we let Y denote the time in minutes between successive commercials, then Y = 60 X. So Var( Y ) = Var(60 X ) = 60 2 Var( X ) = 249 . 75 . 2. A certain art museum has two paintings and two sculptures by Salvador Dal´ ı in its collection. Because of considerations of space, the museum rarely displays all of its Dal´ ı pieces simultaneously, but there is always at least one sculpture on display. Also, the probability that all the pieces are displayed and the probability that exactly one piece is displayed during a given week are both 0 . 1. Let X denote the number of Dal´ ı paintings and Y the number of Dal´ ı sculptures on display during any given week. The marginal probability distributions of X and Y are given below. x
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This homework help was uploaded on 02/11/2008 for the course MATH 218 taught by Professor Haskell during the Fall '06 term at USC.

Page1 / 3

Midterm%201%20Solutions - Midterm 1 Solutions 1 Suppose...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online