Find the cumulative distribution function (cdf) for an ...



Math 309 Test 3 Carter Name________________________

Show all work in order to receive credit.

1. Let [pic]. (20 pts)

a) Find k so that f(x) is a valid probability density function for a random variable X.

b) Find P( X < 0.3)

c) Find P( X < 0.3 | X < 1)

d) Find the mean and variance of X.

2. Emily’s commute to school varies randomly between 22 and 29 minutes. If she leaves at 7:35 a.m. for an 8 a.m. class, what is the probability that she is on time? (7 pts)

3. Use the probability density function to find the cumulative distribution function (cdf) for an exponential random variable with mean [pic]. (7 pts)

4. The number of accidents in a factory can be modeled by a Poisson process averaging 2 accidents per week.

a) Find the probability that the time between successive accidents is more than 1 week.

b) Let W denote the waiting time until three accidents occurs. Find the mean, variance, and the probability density function of W.

c) Find probability that it is less than one week until the occurrence of three accidents. (Setting up the integral is sufficient.) (18 pts)

5. The annual rainfall in a certain region is normally distributed with mean 29.5 inches and standard deviation 2.5 inches.

a) Find the probability that the annual rainfall is between 28 and 29 inches.

b) How many inches of rain is exceeded only 1% of the time? (14 pts)

6. The proportion of pure iron in certain ore samples has a beta distribution, with [pic]and [pic].

a) Find the probability that one of these samples will have more than 40% pure iron.

b) Find the probability that exactly three out of four samples will have more than 40% pure iron. (14 pts)

7. Select one of the following to prove: (10 pts)

a) If X has an exponential distribution with [pic], then P(X > a+b | X > a) = P ( X > b).

b) Let [pic]. If [pic], then Γ(a+1) = aΓ(a).

c) Derive the mean of the Beta distribution

d) Show that [pic] where f(x) is the pdf of the gamma distribution.

|Answers: |

| |

|1. k = 0.5; .0225; .09; 4/3, 2/9; 2. 3/7 3. [pic] |

|4a. Let T = time from occurrence of 1 accident until the next accident. T is exponential w/ lambda = 2, [pic]. |

|4b. W is gamma w/ s = 3 and lambda = 2, [pic] |

|5a) P(28 < X < 29) = P( -.6 < Z ................
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