STAT/MATH 394A
1. Choose a boy at random, and suppose that he comes from a two-children
family. What is the probability that the other child in the family
is a boy? Assume that gender is determined independently, and
that the probability of a boy is 1/2.
2. The internal temperature in a gizmo is a random variable X
with pdf fX (x) = 11(1-x)10, 0 < x <
1 (in appropriate units). The gizmo has a cutoff feature, so that
whenever the temperature exceeds a
cutoff the gizmo turns off (this only affects the output of the
gizmo, not the internal temperature). It is observed that the
gizmo shuts off with probability 10-22. What is its
cutoff number?
3. A certain kind of exam has a pass rate of 65%. A new type of
exam is tried. It yields the same result as the old one for 80%
of those who would have passed the old one, and also for 80% of
those who would have failed. If a person passes the new exam,
what is the probability that she would have passed the old one?
4. Which of the following functions are cumulative distribution functions?
You must give your reason to get credit.
(a) F(x) = ln ( 1 + ex )
(b) F(x) = 2x, 0 < x < 1/4, and x, 1/4 <= x < 1.
(c) F(x) = x/2, 0 < x < 1/4, and x, 1/4 <= x <
1.
5. Twins among humans can either be identical (obtained from one inseminated egg) or non-identical (obtained from two eggs). In the former case, the twins are necessarily of the same sex, whereas in the other case, they may be either of the same or different sex. Assume that identical twins occur with probability r, and non-identical twins with probability 1-r. In addition, each egg yields one or two males (two in the case of identical twins) with probability p, and one or two females with probability 1-p. The sexes of different eggs are determined independently, and independently of whether there is one or two individuals developing. The sample space consists of the following elementary events:
e1 = { two boys, identical twins }
e2 = { two boys, non-identical twins }
e3 = { oldest twin a boy, youngest a girl }
e4 = { oldest twin a girl, youngest a boy }
e5 = { two girls, identical twins }
e6 = { two girls, non-identical twins }
(a) Find P(ei ), i = 1 ,..., 6.
(b) Let A = { oldest twin a boy }, and B = { identical twins }.
Are A and B independent?