STA 401 Final practice questions Fall 2020
Name
1. For each of the statements below, determine whether they are true (T) or false (F).
(1) A and B are two events, if A and B are independent, then A and B¯ are indepen-
dent.
(2) If A and B are independent, then P (A ∪B) = P (A) + P (B).
(3) P (A ∩B ∩ C) = P (B)P (B|C)P (A|B ∩ C)
(4) A function of a random variable is also a random variable.
(5) Suppose that X ∼ Gamma(1, 3) and Y ∼ Gamma(2, 3). The density functions
of X and Y have the same shape.
(6) Values of variances cannot be negative.
(7) A uniform random variable can always be used to model proportions.
(8) Suppose Y1 and y2 are any two jointly random variables, the conditional expec-
tation Y2 given Y1 is a function of Y2
(9) Correlation coefficient is a measure of any relationship between two random vari-
ables.
(10) Suppose that Cov(X,Y ) = −0.4 and Cov(S, T ) = 0.6. The linear relationship
between S and T is stronger than that between X and Y .
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2. A balanced die is tossed six times, and the number on the uppermost face is recorded
each time. What is the probability that the numbers recorded are 1,2,3,4,5, and 6 in
any order.
3. Of the volunteers donating blood in a clinic, 80% have Rhesus (Rh) factor present in
their blood. If five volunteers are randomly selected, what is the probability that at
least one does not have Rh factor?
4. Industrial accidents occur according to a Poisson distribution with an average of three
accidents per month.
(1) Let Y denote the number of accidents during the next three months. Find the
probability mass function of Y .
(2) Find the probability that six or more than six accidents will occur during the next
three month.
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5. Suppose that X ∼ Binomial(24, 0.5) and Y ∼ Poisson(1), and they are independent,
what is the variance of X + 2Y ?
6. A random variable Y has moment generating function (MGF) m(t) = (1− 3t)−1.
(1) Write down this probability density function of Y .
(2) Find P (Y = 1)
7. The PDF of random variable Y is given by
f(y) =

1
2 0 ≤ y ≤ 1
1
3y 1 < y ≤ 2
0 elsewhere
(1) Find the CDF.
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(2) Graph CDF.
8. The joint probability mass function of Y1 and Y2 is given by
p(y1, y2) y1 = 0 y1 = 1 y1 = 2
y2 = 0 1/6 1/12 1/12
y2 = 1 1/3 c 1/6
(1) Find c such that the table provides a valid probability mass function.
(2) Find the marginal probability mass function of Y1.
(3) Find the cumulative distribution function of Y1.
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(4) Find V (Y1)
(5) Are Y1 and Y2 independent?
9. The joint probability density function for Y1 and Y2 is given by
f (y1, y2) =
ky2 0 ≤ y2 ≤ y1 ≤ 10 elsewhere
(1) Find k such that f(y1, y2) is a valid density function.
(2) Find F (0.5, 0.5)
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(3) Find E(Y1/Y2).
(4) Find the marginal probability density function of Y2.
(5) Find the cumulative distribution function of Y2.
(6) Find E(Y2)
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(7) Find the conditional probability density function f(y1|y2)
(8) What are the distribution and parameter(s) of random variable Y1|Y2 = y2
(9) Find P (0.1 < Y1 < 0.8|Y2 = 0.5).
(10) Find P (0.1 < Y1 < 0.8|0 < Y2 < 0.5)
(11) Find E(Y1|Y2 = y2)
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(12) Find V (Y1|Y2 = y2)
(13) Find E(Y1) using E(Y1) = E[E(Y1|Y2)].
(14) Find the 0.4th quantile of Y1.
10. Assume that Y denotes the number of bacteria per cubic centimeter in a particular
liquid and that Y has a Poisson distribution with parameter λ. Further assume that
λ varies from location to location and has a chi-squared distribution with degrees of
freedom ν . If we randomly select a location, what is the
(1) expected number of bacteria per cubic centimeter?
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(2) standard deviation of the number of bacteria per cubic centimeter?
11. Suppose that Y1 ∼ Gamma(3, 3), Y2 ∼ Gamma(2, 3) and Y3 ∼ Gamma(1, 3) are
independent. Find the distribution and parameter(s) of Y1 + Y2 + Y3.
12. Let Y1 and Y2 be two random variables and V (Y1) = 2, V (Y2) = 2, the correlation
coefficient ρ = 0.4.
(1) Find V (Y1 − 2Y2).
(2) Find Cov(Y1 − 2Y2, Y1).
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