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HW6_10_soln

# HW6_10_soln - IE 111 Spring 2009 Homework#7 Solutions...

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IE 111 Spring 2009 Homework #7 Solutions Question 1. Consider the following probability density function (PDF): f X (x) = kx 2 for 0 x 1 f X (x) = 0 otherwise a) Find k so that we have a valid probability density function. 3 3 3 1 1 0 1 0 3 2 = = = = k thus k x k dx x k b) Find P(X 1). Since the domain is [0,1], P(X 1) = 1 c) Find E(X). 4 3 4 3 3 1 0 1 0 4 2 = = x dx xx d) Find b so that P(X b) = 0.9. 0.965489 9 . 0 9 . 0 1 0 3 3 3 ) ( 3 3 3 0 0 3 2 = = = < < = = = x x x x x dx x x F x x Question 2. Consider the following probability density function (PDF): f X (x) = kx for 0 x 1 f X (x) = k for 1 x 2 f X (x) = 0 otherwise a) Find k so that we have a valid probability density function. AREA = (k/2) + k therefore k = 2/3

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b) Find P(X 1). P(X 1) = 1/3 c) Find E(X). 9 11 6 2 8 9 2 6 2 9 2 3 2 3 2 ) ( 2 1 2 1 0 3 1 0 2 1 = - + = + = + = x x xdx xxdx X E d) Find P(X>1.5). = (0.5)(2/3) = 1/3 Question 3 A random variable X~N(5,16). a) Find P(X 10). = Φ ((10-5)/4) = Φ (1.25) = 0.89435 b) Find P(0 X 10). = Φ (1.25) - Φ (-1.25) = 1-2 Φ (-1.25) = 1-2( 0.10565 ) = 0.7887 c) Find k so that P(X>k) = 0.9 P(X<k) = 0.1 Z(0.1) = -1.28155 = (k-5)/4 thus k=-0.12621 d) Find P(X=5). P(X=5) = 0 since X is continuous Question 4 A machine fills cans of coke. The amount of coke that ends up in cans is a random variable with a mean of 12.2 ounces and a standard deviation of 0.2 ounces. a) There are supposed to be 12 ounces in each can. What percentage of cans will be under filled? P(X<12) = Φ ((12.0-12.2)/0.2) = Φ (-1.00) = 0.158655 b) Suppose I can change the mean from 12.2 to any value I want. What value should I
set the mean to in order that 1% of the cans are under filled? Z(0.01) = -2.32635 = (12 - x)/0.2 x = (2.32635)(0.2)+12 = 12.4653 c) Suppose the mean must stay at 12.2, but I can adjust the standard deviation σ . What value of σ will result in 1% of the cans being under filled?

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HW6_10_soln - IE 111 Spring 2009 Homework#7 Solutions...

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