Statistics 510: Notes 7
Statistics 510: Notes 18
Reading: Sections 6.3-6.5
I. Sums of Independent Random Variables (Chapter 6.3)
It is often important to be able to calculate the distribution of [pic]from the distribution of [pic]and [pic]when [pic]and [pic]are independent.
Sum of Discrete Independent Random Variables:
Suppose that [pic]and [pic]are discrete independent random variables. Let [pic]. The probability mass function of [pic]is
[pic]
where the second equality uses the independence of [pic]and [pic].
Example 1: Sums of independent Poisson random variables. Suppose [pic]and [pic]are independent Poisson random variables with respective means [pic]and [pic]. Show that [pic]has a Poisson distribution with mean [pic].
Sums of Continuous Independent Random Variables:
Suppose that [pic]and [pic]are continuous independent random variables. Let [pic]. The CDF of [pic]is
[pic]
[pic]
By differentiating the cdf, we obtain the pdf of [pic]:
[pic]
Example 2: Sum of independent uniform random variables. If [pic]and [pic]are two independent random variables, both uniformly distributed on (0,1), calculate the pdf of [pic].
Sum of independent normal random variables:
Proposition 6.3.2: If [pic]are independent random variables that are normally distributed with respective means and variances [pic]then [pic]is normally distributed with mean [pic]and variance [pic].
The book contains a proof of Proposition 6.3.2 that calculates the distribution of [pic] using the formula for the sum of two independent random variables established above, and then applying induction. We will prove Proposition 6.3.2 in Chapter 7.7 using Moment Generating Functions.
Example 3: A club basketball team will play a 44-game season. Twenty-six of these games are against class A teams and 18 are against class B teams. Suppose that the team will win each against a class A team with probability .4 and will win each game against a class B team with probability .7. Assume also that the results of the different games are independent. Approximate the probability that the team will win 25 games or more.
II. Conditional Distributions (Chapters 6.4-6.5)
(1) The Discrete Case:
Suppose X and Y are discrete random variables. The conditional probability mass function of [pic]given [pic]is the conditional probability distribution of [pic]given [pic]. The conditional probability mass function of X|Y is
[pic]
(this assumes [pic]). This is just the conditional probability of the event [pic]given that [pic].
If X and Y are independent random variables, then the conditional probability mass function is the same as the unconditional one. This follows because if X is independent of Y, then
[pic]
Example 3: In Notes 17, we considered the situation that a fair coin is tossed three times independently. Let X denote the number of heads on the first toss and Y denote the total number of heads.
The joint pmf is given in the following table:
| |Y | | | |
|X |0 |1 |2 |3 |
|0 |1/8 |2/8 |1/8 |0 |
|1 |0 |1/8 |2/8 |1/8 |
What is the conditional probability mass function of X given Y? Are X and Y independent?
(2) Continuous Case
If X and Y have a joint probability density function [pic], then the conditional pdf of X, given that Y=y , is defined for all values of y such that [pic], by
[pic].
To motivate this definition, multiply the left-hand side by dx and the right hand side by [pic]to obtain
[pic]
In other words, for small values of [pic]and [pic], [pic]represents the conditional probability that X is between x and [pic] given that Y is between y and [pic].
The use of conditional densities allows us to define conditional probabilities of events associated with one random variable when we are given the value of a second random variable. That is, if X and Y are jointly continuous, then for any set A,
[pic].
In particular, by letting [pic], we can define the conditional cdf of X given that [pic]by
[pic].
Example 4: Suppose X and Y are two independent random variables, both uniformly distributed on (0,1). Let [pic]. What is the conditional distribution of [pic]given that [pic]? Are [pic]and [pic]independent?
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