Sample Test 2 Prob&Stats 1
1. [20] Let Xhave the p.d.f. f(x) = cejxjfor ๎1๎x < 1(please note the absolute value
sign).
(a) Show that c= 1=(2e๎2).
(b) Find the median of X.
(c) Find the moment generating function M(t)of X. Check M(0).
(d) Find the expected value of X.
2. [20] Use the moment generating function M(t) = e๎(et๎1)(or the cumulant generating
function) of the Poisson distribution to show that ๎=๎and ๎2=๎.
3. [20] Suppose that the probability of a certain item being defective is p= 0:01. Suppose
further that the quality of an item is independent of the quality of the other items. An
inspector selects ten items at random. Let Xequal the number of defective items in the
sample.
(a) How is Xdistributed?
(b) Give the values of E[X]and V ar [X].
(c) Find P(X= 0),P(X๎1), and P(X๎1).
4. [20] The number of hurricanes per year within 100 miles of Boca Raton follows a Poisson
distribution with mean 0.68. What is the probability that in this zone
(a) no hurricane happened in a given year?
(b) exactly 2 were reported?
(c) at least 2 were reported?
(d) at most 2 were reported?
5. [20] Let Xhave an exponential distribution with mean ๎ > 0. Show that
P(X > x +yjX > x) = P(X > y):
1
