Exercises. 1. Shown below is the joint probability distribution for two random variables X and Y. X Y 510 0.12 0.08 0.30 0.20

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

X	Y 510
	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
	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
	0.06 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.

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