- •3.1. Random variables
- •3.2. Probability distributions for Discrete Random
- •Variables
- •3.3. Expected (mean) value and variance for discrete random variables
- •3.3.1. Expected value
- •3.3.2. Variance and standard deviation of discrete random variable
- •3.3.3. Mean and variance of linear function of a random variable
- •3.4. Jointly distributed discrete random variable
- •3.4.1. Conditional probability function
- •3.4.2. Independence of jointly distributed random variables
- •3.4.3. Expected value of the function of jointly distributed
- •3.4.4. Covariance
- •3.5. The binomial distribution
- •3.5.1. Mean and standard deviation of the binomial distribution
- •3.6. The hypergeometric probability distribution
- •3.7.The Poisson probability distribution
3.4.2. 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.
3.4.3. 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.
3.4.4. 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) ; ; ;
; ; c) d) Yes;
2. a) 0.16; b) 0.76; c) ;
d) e) 11/24; f) 1.15;
g) 0.109; h) No; 3. a) 0.04; b) 2.39; c) 1.63; d) 3/13; e) 3/11; f) No;
4. a) ; ; ; ;
b) c) 20/53; d) 0.057.