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Worksheet V: Discrete random variables: joint pmf
(preliminary version)
1. What is meant by the joint pmf, pX,Y(x, y) of two discrete r.v X and Y?
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3. Let the pmf of Y be given by: pY(0) = ½ , pY(1) = 1/8, pY(2) = 1/8, and, pY(3) = ¼.
Find the cumulative distribution function (cdf) of Y.
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7. Define the cdf, FX(x) of a random variable X.
8. What is the Bernoulli random variable with parameter p? Give its pmf. Prove the
normalization property.
9. Define the Binomial random variable with parameters p and n. Give its pmf.
normalization property.
Prove the
10. Define the Geometric random variable with parameter p. Give its pmf. Prove the
normalization property.
11. Define the Poisson random variable with parameter  > 0. Give its pmf. Prove the
normalization property.
12. What is meant by a discrete Uniform random variable with support {a, b} where a and b
are integers?
Give its pmf.
13. Consider two independent coin tosses, each with a ¾ probability of a head, and let X be
the number of heads obtained. Find the pdf of X. Does X fall under one of the named
categories above?
14. A school class of 120 students is driven in 3 buses to a symphonic performance. There
are 36 students in one of the buses, 40 in another, and 44 in the third bus. When the buses
arrive, one of the 120 students is randomly chosen. Let X denote the number of students
on the bus of that randomly chosen student.
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(d) Are these random variables independent?
16. If X is a random variable and g(x) is a real-valued function, then g(X) is a random
variable. Show that E[g(X)] =
 g ( x) p
xsSupp( X )
X
( x)
17. Show that E is a linear operator. Define Variance and SD of X. Prove that
Var(X) = E[X]2 – E[X2] – E[X]2 . What is meant by the “nth moment” of X?-
18. Consider the Bernoulli, Uniform, Geometric, Binomial, and Poisson RVs. Verify the
normalization property. Compute the mean and variance of each RV.
19. What is meant by the joint pmf, pX,Y(x, y) of two discrete r.v X and Y?
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21. A fair coin is tossed repeatedly and independently until 2 consecutive heads or 2
consecutive tails appear. Find the PMF, the expected value, and the variance of the
number of tosses.
22. Consider two independent coin tosses, each with a 3/4 probability of a head, and let X
be the number of heads obtained. This is a binomial random variable with parameters n =
2 and p = 3/4. Find E[X].
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25. Interpret the following diagram: