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SCHOOL OF ENGINEERING SCIENCES FAEN 302 : STATISTICS FOR ENGINEERS SECOND SEMESTER 2014/15 EXERCISE 2 1. Let X be a random with probability density function variable c(1 − x2 ), −1 < x < 1 f (x) = 0, otherwise
(a) What is the value of c? (b) What is the cumulative distribution function of X? 2. If you buy a lottery ticket in 50 lotteries, in each of which your chance of 1 , what is the (approximate) probability that you will winning a prize is 100 win a prize (a) exactly once? (b) at least twice? 3. The speed of a molecule in a uniform gas at equilibrium is a random variable whose probability density function is given by f (x) =
2
ax2 e−bx , x ≥ 0 0, x < 0
where b = m/2kT and k, T, and m denote, respectively, Boltzmann’s constant, the absolute temperature of the gas, and the mass of the molecule. Evaluate a in terms of b. 4. If X is uniformly distributed over (-1, 1), find (a) P {|X| > 21 } (b) the density function of the random variable |X|. 5. The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter λ = 12 . What is (a) the probability that a repair time exceeds 2 hours? (b) the conditional probability that a repair takes at least 10 hours, given that its duration exceeds 9 hours? 6. The lifetime in hours of an electronic tube is a ran- dom variable having a probability density function given by, f (x) = xe−x , x ≥ 0 Compute the expected lifetime of such a tube.
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7. Buses arrive at a specified stop at 15-minute intervals starting at 7 A.M. That is, they arrive at 7, 7:15, 7:30, 7:45, and so on. If a passenger arrives at the bus stop at a time that is uniformly distributed between 7 and 7:30, find the probability that he waits (a) less than 5 minutes for a bus. (b) more than 10 minutes for a bus. 8. Suppose that the random variable X is equal to the number of hits obtained by a certain baseball player in his next 3 at bats. If P {X = 1} = .3, P {X = 2} = .2, and P {X = 0} = 3P {X = 3}, nd E[X]. 9. A purchaser of electrical components buys them in lots of size 10. It is his policy to inspect 3 components randomly from a lot and to accept the lot only if all 3 are nondefective. If 30 percent of the lots have 4 defective components and 70 percent have only 1, what proportion of lots does the purchaser reject? 10. Life expectancy (in days) of electronic component has CDF F (x) =
0, x < 1 1− x≥1 1 x,
(a) Find the pdf of X. (b) Find the probability that the component lasts for more than 10 days. 11. Someone claims the figure below is the CDF for grades on the FAEN 302 final exam and that the probability that a random student scored 50 or lower is 0.5. Evaluate the claim being made.
12. A system consisting of one original unit plus a spare can function for a random amount of time X. If the density of X is given (in units of months) by cxe−x/2 , x > 0 f (x) = 0, x ≤ 0 what is the probability that the system functions for at least 5 months? 2
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13. Consider a jury trial in which it takes 8 of the 12 jurors to convict the defendant; that is, in order for the defendant to be convicted, at least 8 of the jurors must vote him guilty. If we assume that jurors act independently and that, whether or not the defendant is guilty, each makes the right decision with probability θ, what is the probability that the jury renders a correct decision?
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