1. Suppose a coin is flipped 17 times independently. The probability of getting
heads on any one coin toss is .4.
(a) What is the probability that 10 of the 17 coin tosses come up tails?
(b) Compute the probability that at least 2 coin tosses come up heads.
(c) If X= number of heads among the 17 flips, compute the expected value of
X.
(d) If Y = number of tails among the 17 flips, compute V ar(Y ).
2. Suppose that every time little Billy goes up to bat for his little league team,
the probability that he gets on base is .45. Also, the result of any one at-bat is
independent of all of the others. Although Billy is a great baseball player, he
loathes the game, and decides he will quit playing for the rest of his life after
he gets on base 4 more times.
(a) What is the probability that it takes 7 at-bats for Billy to get on base 4
times?
(b) What is the probability that it will take more than 6 at-bats for Billy to
get on base 4 more times.
(c) What is the expected number of outs (at-bats where he doesn’t get on
base) that it will take Billy before he quits baseball?
3. Suppose you are playing a lottery with other players in which you select 7
numbers between 1 and 50, and then balls are drawn out of a bin which has
50 balls numbered 1 through 50. The first player to have all of their numbers
drawn wins. Suppose you have made a selection of 7 numbers.
(a) What is the probability that all of your numbers are drawn by the tenth
draw?
(b) What is the probability that all of your numbers are drawn by the twentieth
draw?
(c) How many of your numbers do you expect to have seen by the fifteenth
draw?
4. Suppose that
(
f (x) =
2e−cx : 0 ≤ x < ∞
0
: otherwise
is the pdf of a continuous random variable X.
(a) Compute c so that f (x) is a valid pdf.
(b) Compute the probability that 3 ≤ X < 4
(c) Compute the probability that X > 1
(d) Compute the 75th percentile of X.
(e) Compute E(X).
(f) Compute E(4X + 2)
(g) Compute the CDF of X.
5. Suppose that
(
f (x) =
x −x2 /(2θ2 )
e
θ2
0
:0≤x<∞
: otherwise
is the pdf of a continuous random variable X. θ is assumed to be a positive
constant.
(a) Compute the CDF of X.
(b) Compute the probability that 4 ≤ X ≤ 7 if θ = 2.
6. Let X denote the amount of time for which a book on 2-hour reserve at the
Marriot Library is checked out by a randomly selected student and suppose that
X has CDF given by

:x<0

 02
x
F (x) =  4 : 0 ≤ x < 2

1
:2≤x
(a) What is the probability that X is less than 1?
(b) Compute the pdf of X, f (x).
(c) Compute the 20th percentile X.
(d) Compute E(X) and V ar(X).
(e) Compute E[(3X − 2)2 ]
7. Compute the 30th percentile of a U nif (4, 9) random variable.
8. Compute the probability that a standard normal random variable has an absolute value larger than 1.5. i.e. Compute
P {|N (0, 1)| > 1.5}
9. Suppose that the heights of 13 year old boys follow a normal distribution with
mean 4.73 feet and standard deviation .25 feet. What is the probability that a
randomly selected 13 y.o. boy is taller than 5 feet?
10. Girls heights at age 13 also follow a normal distribution, but with mean 4.75
feet and standard deviation .20 feet. Morgan, who is a girl and is currently
10 years old, is a basketball enthusiast and wants to be taller than 75% of the
other girls. How tall must Morgan be once she turns 13 so that the probability
that she is taller than a randomly selected girl is .75?
11. Compute c so that P (N (10, 52 ) > c) = .12
12. Suppose that X and Y are jointly distributed, continuous RVs with joint density
(
f (x, y) =
24xy : 0 ≤ x, y ≤ 1 and x + y ≤ 1
0
: otherwise
(a) Compute the probability that X ≤ Y .
(b) Compute the marginal density of X.
(c) Compute the marginal density of Y .
13. Winston runs a small soup and salad cafe. If a random customer enters Winston’s shop, and if X= amount of money the customer spends on soup and Y =
amount of money the customer spends on salad, then X and Y have joint pmf:
p(x,y) 2
Y
2
.1
3
.05
X
3
4
.3 .2
.25 .1
(a) Compute the probability that X is larger than Y .
(b) Compute the probability that the total amount that a random customer
spends is over 5.
(c) Compute the marginal pmf of X.
(d) Compute the marginal pmf of Y .