statistic

1. If the m.g.f. of a random variable X is ψ(t) = et2 for −∞ < t < ∞, what is the distribution of X?

question about the normal distribution.

2. If a random sample of 25 observations is taken from the normal distribution with mean μ and standard deviation 2, what is the probability that the sample mean will lie within one unit of μ?

question about the normal distribution.

3. Suppose that the random variables X1, . . . , Xk are in- dependent and Xi has the exponential distribution with parameter βi (i =1,...,k). Let Y =min{X1,...,Xk}. Show that Y has the exponential distribution with param- eter β1 + . . . + βk.

Questions about the Gamma distribution

HINT: what’s the opposite of the minimum being less than or equal to t?

4. Suppose that a certain examination is to be taken by five students independently of one another, and the num- ber of minutes required by any particular student to com- plete the examination has the exponential distribution for which the mean is 80. Suppose that the examination be- gins at 9:00 a.m. Suppose that the examination considered is taken by five students, and the first student to complete the examination finishes at 9:25 a.m. Determine the probability that at least one other student will complete the examination before 10:00 a.m.

Questions about the Gamma distribution

5. Suppose that two different tests A and B are to be given to a student chosen at random from a certain population. Suppose also that the mean score on test A is 85, and the standard deviation is 10; the mean score on test B is 90, and the standard deviation is 16; the scores on the two tests have a bivariate normal distribution; and the correlation of the two scores is 0.8. If the student’s score on test A is 80, what is the probability that her score on test B will be higher than 90?

Question about The Bivariate Normal Distributions