Download Quiz-5 solutions

Probability, 2014 Fall
Quiz #5, solutions
Let p(x,y) be a joint probability mass function (pmf) of discrete random variables X and Y.
Answer the following questions as complete as possible.
1. What are the conditions for p(x,y) to be a joint pmf?
Answer:
1) 𝑝(𝑥, 𝑦) ≥ 0
2) ∑𝑥∈𝐴 ∑𝑦∈ 𝐵 𝑝(𝑥, 𝑦) = 1
2. What is the marginal pmf of X?
You should also give the notation and the formula of definition.
Answer: X’s marginal probability mass function is 𝑝𝑥 (𝑥) = ∑𝑦∈𝐵 𝑝(𝑥, 𝑦). Basically given the X and Y
as two random variables in addition to their joint distribution, the marginal distribution of X is the
probability distribution of X when the value of Y is not cared. As one can see in the formula it is
calculated by summing the joint probability distribution over Y.
3. What is the joint (cumulative) distribution function (cdf) of X and Y?
You should also give the notation and the formula of definition.
Answer: 𝐹(𝑥, 𝑦) = 𝑃(𝑋 ≤ 𝑥, 𝑌 ≤ 𝑦) = ∑𝑥𝑖 ≤𝑥 ∑𝑦𝑖 ≤𝑦 𝑝(𝑥𝑖 , 𝑦𝑖 )
The joint cumulative distribution function gives the accumulated probability that X is up to x and Y is up
to y. In addition we have 0 ≤ 𝐹(𝑥, 𝑦) ≤ 1 for all x,y.
4. What is the marginal cdf of X?
You should also give the notation and the formula of definition.
Answer: X’s marginal probability density functions is
𝐹𝑋 (𝑡) = 𝑃(𝑋 ≤ 𝑡) = 𝑃(𝑋 ≤ 𝑡, 𝑌 ≤ ∞) = 𝐹(𝑡, ∞)
This is the accumulated probability of X up to t when the values of Y are not cared.
5. What is the conditional probability function of X, given that Y=y?
You should also give the notation and the formula.
Answer: Let X and Y be two discrete random variables. The conditional probability function of X
given that Y=y is
𝑃(𝑋 = 𝑥, 𝑌 = 𝑦) 𝑝(𝑥, 𝑦)
𝑝𝑋|𝑌 (𝑥|𝑦) = 𝑃(𝑋 = 𝑥 |𝑌 = 𝑦) =
=
𝑃(𝑌 = 𝑦)
𝑝𝑌 (𝑦)
Where 𝑥 ∈ 𝐴, 𝑦 𝑖 ∈ 𝐵 and 𝑝𝑌 (𝑦) > 0. It is the probability of X when the value of Y is fixed to y.
1
6. Give the definition for E(X|Y=y)?
Answer: The conditional expectation of the random variable X given that Y=y is
𝐸(𝑋|𝑌 = 𝑦) = ∑ 𝑥𝑝𝑋|𝑌 (𝑥|𝑦)
𝑥∈𝐴
Where 𝑝𝑌 (𝑦) > 0.
2