Suppose that a pair of random variables have the same ...



STAT 211

Handout 5 (Chapter 5)

Joint Probability Distributions and Random Samples

By using the following examples, the joint probability mass function for two discrete random variables and their properties, their marginal probability mass functions, the case for independent and dependent variables, their conditional distributions, expected value, variance, covariance, and correlation will be demonstrated. Properties will be explained.

1. Quality audit records are kept on numbers of major and minor failures of circuit packs during-in of large electronic switching devices. They indicate that for a device of this type, the random variables X (the number of major failures) and Y (the number of minor failures) can be described at least approximately by the accompanying joint distribution.

| x|0 |1 |2 |

|y | | | |

|0 |0.15 |0.05 |0.01 |

|1 |0.10 |0.10 |0.04 |

|2 |0.10 |0.14 |0.04 |

|3 |0.10 |0.13 |0.04 |

a. Find the marginal probability functions for X and Y.

b. Are X and Y independent?

c. Find the expected value and the variance of X.

d. Find the expected value and the variance of Y.

e. Find the Cov(X,Y) and Corr(X,Y).

f. Find the conditional distribution for Y given that X=0 that is there are no circuit pack failures.

g. What is the mean for the conditional distribution of Y given that X=0?

h. Suppose that demerits are assigned to devices of this type according to the formula D=5X+Y. Find the expected value of D.

i. Suppose that demerits are assigned to devices of this type according to the formula D=5X+Y. Find the variance of D.

j. Suppose that demerits are assigned to devices of this type according to the formula U=Min(X,Y). Find the mean value and the variance of U.

By using the following example, the joint probability density function for two continuous random variables and their properties, their marginal probability density functions, the case for independent and dependent variables, their conditional distributions, expected value, variance, covariance, and correlation will be demonstrated. Properties will be explained.

2. Suppose that a pair of random variables, X and Y have the same joint probability density

[pic]

a. Find the marginal probability density functions for X and Y.

b. Are X and Y independent?

c. Evaluate P(X+2Y(1).

d. Find the expected value and variance of X.

e. Find the expected value and variance of Y.

f. Find the conditional probability function of x given y=0.6.

g. Find the conditional probability function of y given x=0.4.

h. What is E(X|Y=0.6)?

i. Find the Cox(X,Y) and Corr(X,Y).

Random Sample:

The random variables X1, X2, ….,Xn are said to form a random sample of size n if

(i) The Xi's are independent random variables.

(ii) Every Xi's has the same probability distribution.

The sampling distribution of [pic]: Discussion theoretically

Exercise 5.42 (textbook): A company maintains 3 offices in a certain region, each staffed by two employees. Information concerning yearly salaries (1000’s of dollars) is as follows:

Office 1 1 2 2 3 3

Employee 1 2 3 4 5 6

Salary 19.7 23.6 20.2 23.6 15.8 19.7

a) Suppose two of these employees are randomly selected from among the six (without replacement). Determine the sampling distribution of the sample mean salary [pic].

S={(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,4),(3,5),(3,6), (4,1),(4,2),(4,3),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5)}. There are 30 outcomes.

[pic]

65. when (1,2),(2,1),(1,4),(4,1),(2,6),(6,2),(4,6),(6,4) with 8 outcomes

95. when (1,3),(3,1),(3,6),(6,3) with 4 outcomes

75. when (1,5),(5,1),(5,6),(6,5) with 4 outcomes

19.70 when (1,6),(6,1),(4,5),(5,4),(2,5),(5,2) with 6 outcomes

90. when (2,3),(3,2),(3,4),(4,3) with 4 outcomes

60. when (2,4),(4,2) with 2 outcomes

18. when (3,5),(5,3) with 2 outcomes

[pic] : 17.75 18 19.7 19.95 21.65 21.90 23.60 otherwise

[pic] : 4/30 2/30 6/30 4/30 8/30 4/30 2/30 0

E([pic])=[pic]=613/30=20.43

b) Suppose one of the three offices is randomly selected. Determine the sampling distribution of the sample mean salary [pic] .

S={(1,2),(2,1),(3,4),(4,3),(5,6),(6,5)}. There are 6 outcomes

[pic]

21.65 when (1,2),(2,1) with probability 1/3

21.90 when (3,4),(4,3) with probability 1/3

17.75 when (5,6),(6,5) with probability 1/3

E([pic])=61.3/3=20.43

Population mean = (19.7+23.6+20.2+23.6+15.8+19.7)/6=20.43

Additional: The sampling distribution of the range of salaries for part (b) in the previous question is

range: 3.4 3.9

p(range) : 2/6 4/6

E(range)= [pic]=3.7333 where Population range is 23.6-15.8=7.8

The distribution of a linear combination of variables: Discussion theoretically

Exercise 5.60:

First two cars are economy brand and distributed N(20,4)

Last three cars are name brand and distributed N(21,3.5)

All five are independent.

Y is a measure of the difference in efficiency between economy gas and name brand gas.

E(Y)= [pic]=[pic]=20-21=-1

Var(Y)= [pic]=[pic]=2+1.6667=3.6667

P(Y(0)=P(Z(0.52)=0.3015

P(-1(Y(1)=P(Y(1)-P(Y ................
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