Statistics 510: Notes 7 - Statistics Department
Statistics 510: Notes 19
Reading: Sections 6.3, 6.4, 6.5, 6.7
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 X and Y when X and Y are independent. At the end of last class, we derived the results:
[pic]
and
[pic]
Example 1: Sum of two independent uniform random variables. If X and Y are two independent random variables, both uniformly distributed on (0,1), calculate the pdf of [pic].
II. Conditional Distributions (Chapters 6.4-6.5)
(1) The Discrete Case:
If X and Y are jointly distributed discrete random variables, the conditional probability that [pic]given that [pic]is, if [pic], then the conditional probability mass function of X|Y is
[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].
Note that we have been able to give workable expressions for conditional probabilities even though the event on which we are conditioning (namely the event [pic]) has probability zero.
Example 4: Suppose X and Y are two independent random variables, both uniformly distributed on (0,1). Let [pic](these are called the order statistics of the sample – Section 6.6). What is the conditional distribution of [pic]given that [pic]? Are [pic]and [pic]independent?
III. Joint Probability Distribution of Functions of Random Variables
Let [pic]and [pic]be jointly continuous random variables with joint pdf [pic]. It is sometimes of interest to obtain the joint distribution of random variables [pic]and [pic], which arise as functions of [pic]and [pic]. Specifically, suppose that [pic]and [pic]for some functions [pic]and [pic].
Assume that the functions [pic]and [pic]satisfy the following conditions:
1. The equations [pic]and [pic]can be uniquely solved for [pic]and [pic] with solutions given by, say, [pic].
2. The functions [pic]and [pic]have continuous partial derivatives at all points [pic]and are such that the following 2x2 determinant
[pic]
at all points [pic].
Under these two conditions, it can be shown that the random variables [pic]and [pic]are jointly continuous with joint density function given by
[pic] (1.1)
A proof of equation (1.1) proceeds along the following lines:
[pic]
The joint density function can now be obtained by differentiating the above equation with respect to [pic]and [pic]. That the result of this differentiation will be equal to the right hand side of equation (1.1) is an advanced calculus result.
Example 5: Let [pic]denote a random point in the plane and assume that the rectangular coordinates X and Y are independent standard normal random variables. We are interested in the joint distribution of [pic], the polar coordinate representation of the point. ([pic])
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