1. Suppose X and Y are jointly distributed random variables with joint density
function
(
c(x + y 2 ) : 0 ≤ x, y ≤ 1
f (x, y) =
0
: otherwise
(a) Compute c so that f (x, y) is a valid joint density function.
(b) Compute the probability that X + Y > .5.
(c) Compute the marginal density of X
(d) Compute Cov(X, Y ). Are X and Y independent? (Recall Cov(X, Y ) =
E(XY ) − E(X)E(Y ))
2. If a random customer enters the Smiths Grocery on 8th and 9th let X denote
number of peaches purchased by the customer and Y denote number of oranges
purchased by the customer. The management of the Smiths has estimated that
the joint pmf of X and Y is given by the following table.
X
1
2
p(x,y) 0
Y
1
.2 .3 .1
2
.25 .13 .02
(a) Use this joint pmf to compute the probability that the difference between
the number of peaches purchased and the number of oranges purchased by
a random customer is larger than or equal to 1.
(b) Compute E(X + Y )
3. 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 .
(d) Determine whether or not X and Y are independent.
(e) Compute E(XY ).
4. 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:
X
p(x,y) 2
3
4
Y
2
.1 .3 .2
3
.05 .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 .
(e) Are X and Y independent?
(f) Compute E(X + Y ).
5. Suppose that X1 , ..., X10 is a random sample from a Normal Distribution with
mean 7 and variance 15.
(a) Compute the probability that X̄ is larger than 7.5.
(b) What is the probability that X1 + ... + X10 is between 60 and 80?
6. Suppose X1 , ..., X25 is random sample of size 25 from a population whose distribution has probability density function:
(
√
x :0≤x< 2
f (x) =
0
: otherwise
(a) What is the approximate distribution of the sample mean X̄?
√
(b) Compute the approximate probability that X̄ is less than
3
2
(c) Discuss the validity of the approximation you used in parts (a) and (b).
7. Suppose that 4 identical quadruplets are running a relay race. The distribution
of each individual quadruplet’s time for their respective leg of the race is an iid
N (30sec, 9sec2 ). What is the probability that their total time for the relay is
less than the record time of 110 seconds?
8. Suppose X1 , ..., X49 is a sample from the distribution whose CDF is given in
problem 6.
(a) What is the approximate probability that X̄ is larger than 1.5?
(b) What is the approximate probability that X1 + ... + X49 is less than 150?
9. Suppose X1 , ..., X49 is a sample from a Normal Distribution with mean µ and
variance 9 where µ is unknown. Derive a 98% confidence interval for µ using
the statistic X̄.
10. An insurance company is reviewing its current rates. When the rates were originally determined it was believed that the average amount of a claim was $1800.
The owners of the company are concerned that the actual average amount of
a claim is higher than this. The accountants at the company randomly select
50 claims and calculate a sample mean of $1975. Assume that the standard
deviation of claims is $525.
(a) Determine appropriate hypotheses for a hypothesis test in this situation.
(b) Perform the hypothesis test devised above at significance level .1.
(c) Were any approximations used to perform this test? If yes, explain whether
or not they are valid.
(d) Compute a 90% confidence interval for the true average claim based on
this sample.
11. The university claims that it takes on average 20 minutes to find a parking spot
on campus. A random sample of 7 students were asked how long it took them
to find a parking spot. Their responses are displayed in the table below. It is
assumed that the amount of time required to find a spot is normally distributed
(N (µ, σ 2 )).
Student
1
2
3
4
5
6
7
time required to find spot (min) 19.6 15 21.5 20.2 17.8 18.3 19.1
(a) Test the hypotheses
H0 : µ = 20
versus
HA : µ < 20
at significance level .05 based on this sample.
(b) Compute a 99% confidence interval for µ based on the sample.
12. Suppose x1 , ..., x49 is a sample from a N (µ, σ 2 ) distribution where µ, σ are unknown. x̄ and S were computed from the sample, x̄ = 14 and S = 6.4.
(a) Using this sample, test at level .01
H0 : µ = 16
versus
HA : µ < 16
(b) Derive a 98% confidence interval for µ based on the sample.
(c) Derive a 95% confidence interval for σ 2 based on the sample.
13. A die is rolled 500 times, and 94 of those times the outcome was a 3. Suppose
p = probability of getting a 3.
(a) Using this sample, test at level .01
H0 : p = 61
versus
HA : p > 16
(b) Derive a 95% confidence interval for p.
(c) Discuss any approximations that where used above and their validity.
14. Tillamook County Creamery performs a survey on 1000 randomly selected customers, of which 343 preferred vanilla ice cream over all other flavors. Suppose
that p is the true proportion of the population which prefers vanilla ice cream.
(a) Using this sample, test at level .05
H0 : p = .35
versus
HA : p 6= .35
(b) Derive a 95% confidence interval for p.