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Unformatted text preview: Homework 1 (due Jan. 29) ^ 1. Consider one 70% shooter attempting 20 free throws. Let H be the number of free throws made by him. ^ a) Find the probability distribution of H. ^ b) What is the expectation of H? c) What is its standard deviation? 2. A lifetime of a light bulb is normally distributed with the mean of 10000 hours and standard deviation of 1000 hours. There are 100 such bulbs in the building (installed at the same time). a) What is the probability that any given bulb lasts more than 12000 hours? b) What is the probability that any given bulb burns out before 9000 hours? c) What is the probability that we would have to replace at least one bulb in 5000 hours? d) Assuming we do not replace bulbs, what is the probability that the building is in total darkness in 15000 hours? 3. A pharmaceutical company sells serum in vials with nominal volume of 2.5 ml. A regulation specifies that no more than 1% of all vials should be under-filled. The machine filling the vials does so with a standard deviation of 0.1 ml. How much serum will the company have to put in a vial on average in order to meet the regulation? Assume that the volume of serum is normally distributed. 4. Which of the following functions can not represent a probability density functions for any distribution? if x < 0 a) f (x) = 0 if x > 1 -1 if 0 < x < 1 0 1 b) f (x) = 0 if x < 0 if x > 10/11 c) f (x) = 0 1.1 if 0 < x < 10/11 d) f (x) = e) f (x) = f) f (x) = g) f (x) = 0 0 if x < 0 0 if x > 1 1 if 0 < x < 1 if x < 0 0 if x > 1 1.1 if 0 < x < 1 0 if x < 0 5 exp(-5x) if x > 0 0 if x < 0 5(exp(-5x) - 0.1) if x > 0
1 2 exp(-x2 ) 5. Find the value of C such that f (x) is a probability density function for some distribution. 0 if x < 0 a) f (x) = C exp(-x) if x > 0 b) f (x) = C exp(-|x|) if x < 0 c) f (x) = Cx if 0 < x < 3 0 if x > 3 0 2 ...
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This note was uploaded on 03/17/2008 for the course IE 121 taught by Professor Perevalov during the Spring '08 term at Lehigh University .
- Spring '08