Conditional probability function

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Let X and Y be a pair of jointly distributed discrete random variables. The conditional probability function of the random variable Y, given that the random variable X takes the value x, expresses the probability that Y takes the value y, as a function of y, when the value x is specified for X.

This is denoted by , and defined as

Similarly, the conditional probability function of X, given Y=y is

For example, using the table 3.8, we can compute the conditional probability of y=2, given that x=1 as

It means, the probability that randomly chosen student who has 1 test eats 2 snacks is 1/7.

The probability of , given that is

It means, the probability that randomly chosen student who eats 3 snacks has 2 tests is 11/26.

Independence of jointly distributed random variables

Definition:

Let X and Y be a pair of jointly distributed discrete random variables. They are said to be independent if and only if their joint probability function is the product of their marginal probability functions:

for all possible pairs of values x and y. Otherwise they are said to be dependent.

As an example, from table 3.8, let .

Then

; ; .

so number of eaten snacks and number of tests are not independent.

Expected value of the function of jointly distributed

Random variables

Let X and Y be a pair of discrete random variables with probability function .

The mean of random variable X is

The mean of random variable Y is

The mean, or expectation of any function of the random variables X and Y is defined as:

As an example let us calculate means of X, Y, and g(X, Y) for the Example above.

The mean of X is:

It means, on average we expect that each student eats 1.1 snacks per day during final examination week.

The mean of Y is:

It means, on average, we expect that each student has 1.55 tests per day during final examination week.

Covariance

Suppose that X and Y are pair of random variables and they are dependent. We use covariance to measure the nature and strength of the relationship between them.

Definition:

Let X be a random variable with mean , and let Y be a random variable with mean .The expected value of is called the covariance between X and Y, denoted , defined as

An equivalent expression for is:

If is a positive, then there is a positive linear association between X and Y, if is a negative value, then there is a negative linear association between X and Y. An expectation of 0 for would imply an absence of linear association between X and Y.

Let us calculate for probability distribution shown in the

table 3.8.

Using an equivalent expression for yields:

It means that there is a weak negative association between number of tests taken a day during a final examination week and number of eaten snacks.

Exercises

1. Shown below is the joint probability distribution for two random variables X and Y.

X	Y 5 10
10 20 30	0.12 0.08 0.30 0.20 0.18 0.12	0.20 0.50 0.30
	0.60 0.40	1.00

a) Find , , and .

b) Specify the marginal probability distributions for X and Y.

c) Compute the mean and variance for X and Y.

d) Are X and Y independent random variables? Justify your

answer.

2. There is a relationship between the number of lines in a newspaper advertisement for an apartment and the volume of interest from the potential renters. Let volume of interest be denoted by the random variable X, with the value 0 for little interest, 1 for moderate interest, and 2 for heavy interest. Let Y be the number of lines in a newspaper. Their joint probabilities are shown in the table

Number of lines (Y)	Volume of interest (X) 0 1 2
3 4 5	0.09 0.14 0.07 0.07 0.23 0.16 0.03 0.10 0.11

a) Find and interpret .

b) Find the joint cumulative probability function at X=2, Y=4,

and interpret your result.

c) Find and interpret the conditional probability function for Y,

given X=0.

d) Find and interpret the conditional probability function for X,

given Y=4.

e) If the randomly selected advertisement contains 5 lines, what is the probability that it has heavy interest from the potential renters?

f) Find expected number of volume of interest.

g) Find and interpret covariance between X and Y.

h) Are the number of lines in the advertisement and volume of interest independent of one another?

3. Students at a university were classified according to the years at the university (X) and number of visits to a museum in the last year.

(Y=0 for no visits, 1 for one visit, 2 for two visits, 3 for more than two visits). The accompanying table shows joint probabilities.

Number of visits (Y)	Years at the university (X) 1 2 3 4
0 1 2 3	0.06 0.08 0.07 0.02 0.08 0.07 0.06 0.01 0.05 0.05 0.12 0.02 0.03 0.06 0.18 0.04

a) Find and interpret

b) Find and interpret the mean number of X.

c) Find and interpret the mean number of Y.

d) If the randomly selected student is a year student, what is the probability that he or she) visits museum at least 3 times?

e) If the randomly selected student has 1 visit to a museum, what is the probability that he (or she) is a year student?

f) Are number of visits to a museum and years at the university independent of each other?

4. It was found that 20% of all people both watched the show regularly and could correctly identify the advertised product. Also, 27% of all people regularly watched the show and 53% of all people could correctly identify the advertised product. Define a pair of random variables as follows:

X=1 if regularly watch the show; X=0 otherwise

Y=1 if product correctly identified; Y=0 otherwise.

a) Find the joint probability function of X and Y.

b) Find the conditional probability function of Y, given X=0.

c) If randomly selected person could identify the product correctly, what is the probability that he (or she) regularly watch the show?

d) Find and interpret the covariance between X and Y.

Answers

1. a) 0.08; 0.18; 0.30; b) ; ; ;