Unit ii rpq

UNIT – II

TWO DIMENSIONAL RANDOM VARIABLES

PART – A

1. Define joint probability density function of two random variables and .

If is a two dimensional continuous random variable such that

, then

is called the joint pdf of , provided satisfies the following conditions

2. State the basic properties of joint distribution of where and are random variables.

Properties of joint distribution of are

3. Can the joint distributions of two random variables and be got if their marginal distributions are random?

If the random variables and are independent, then the joint distributions of two random variables can be got if their marginal distributions are known. 4. Let and be two discrete random variable with joint pmf

. Find the marginal

pmf of and . The joint pmf of is given by

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X Y

1 2

1

2

Marginal pmf of is

5. Let and be integer valued random variables with and . Are

independent? The marginal pmf of is

The marginal pmf of is

Therefore are independent random variables.

6. The joint probability density function of the random variable is given by

. Find the value of k. Given is the joint pdf , we have put

10 0

22

dydxeyxk yx

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put

Therefore, the value of is .

7. The joint pdf of the random variable is .

Find the value of . Given is the joint pdf , we have

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8. The joint pdf of the random variable is

. Find the value of . Given is the joint pdf , we have

Therefore the value of is .

9. If two random variables have probability density function for . Evaluate . Given is the joint pdf , we have

10. If the function is to be a density function, find the value of c.

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Given is the joint pdf , we have

Therefore the value of is

11. Find the marginal density functions of if

.

Marginal density of is


12. If have joint pdf . Check whether are independent.

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Therefore, are not independent variables.

13. If are random variables having the joint density function

, find .

.

14. Let be continuous random variable with joint pdf

. Find .

.

15. If the joint pdf of is given by , find .

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.

16. Let be random variable with joint density function . Find .

17. Let be any two random variables and be constants. Prove that .

.

18. If , find . .

19. If has mean and variance while has mean and variance and the two are independent, find . Given .20. Find the acute angle between the two lines of regression. The equations of the regression lines are

Slope of line is

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Slope of line is

If is the acute angle between the two lines, then

.

21. If are random variables such that where and are real

constants, show that the correlation co-efficient between them has magnitude one.

Correlation co-efficient

.

Therefore

and

Therefore, the correlation co-efficient between them has magnitude one.22. State central limit theorem. If be a sequence of independent identically distributed random variables with and , and if

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, then under certain general conditions, follows a normal distribution with mean and variance as n tends to infinity.

23. Write the applications of central limit theorem.(i) Central limit theorem provides a simple method for

computing approximate probabilities of sums of independent random variables.

(ii) It also gives us the wonderful fact that the empirical frequencies of so many natural “populations” exhibit a bell shaped curve.

PART - B

1. Three balls are drawn at random without replacement from a box containing 2 white, 3 red and 4 black balls. If denotes the number of white balls drawn and denotes the number of red balls drawn, find the joint probability distribution of .Solution: As there are only 2 white balls in the box, can take the values 0, 1 and 2 and can take the values 0, 1, 2 and 3 since there are only 3 red balls.

[ since only 3 balls are drawn ]

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The joint probability distribution of may be represented in the form of a table as given below

YX

0 1 2 3

0

1 0

2 0 0

2. The joint probability mass function of is given by

, , . Find all the marginal and conditional probability distributions. Also find the probability distribution of .Solution:The joint probability distribution of is given below

Y

X1 2 3

0 3k 6k 9k

1 5k 8k 11k

2 7k 10k 13k

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Since is a probability mass function, we have

Marginal probability distribution of

Marginal probability distribution of

Conditional distribution of given 1Y

Conditional distribution of given



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Probability distribution of

1

2

3

4

5

3.The joint distribution of where and are discrete is given in the following table Solution:

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Verify whether and are independent. Marginal distribution of is Marginal distribution of is

and are independent if for all and (ie) We have to show that

------------------(1) -------------------------------(2)From (1) and (2), we have

Therefore for all and , Hence, the random variables and are independent.

4. The joint pdf of a two dimensional random variable is given

by . Compute (1) (2)

(3) (4) (5) and (6)

. Solution :

(1)

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YX

0 1 2

0 0.1 0.04 0.06

1 0.2 0.08 0.12

2 0.2 0.08 0.12

(2)

(3)

(4)

(5)

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(6)

5. Given the joint pdf of . Find the marginal densities of and . Are and independent? Solution:



Therefore and are independent.

6. The joint pdf of a two dimensional random variable is given by

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. Find the value of and marginal and conditional density functions. Solution: Given is the joint pdf, we have

Therefore,



Conditional density of given is

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.

7. Given the joint pdf of as . Find the marginal and conditional probability density functions of and . Are and

are independent? Solution: Marginal density of is


Therefore and are not independent.



8. Given .(1) Evaluate c, find (2) (3) and (4) . Solution:(1) Given is the joint pdf, we have

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Therefore ,

(2)

(3)

(4)

9. Suppose the pdf of is given by

. Obtain the marginal pdf of

, the conditional pdf of given and then . Solution: Marginal density of is


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10. Find for the following discrete bivariate distribution

Solution:


Marginal distribution of is Marginal distribution of is

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XY

5 15

10 0.2 0.4

20 0.3 0.1


.

11. If is the joint pdf of the random variables and , find the correlation co-efficient of and . Solution:


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12. Find the correlation between and if the joint probability density of and is . Solution:


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13. If the independent random variables and have the variances and

respectively, find the correlation co-efficient between and .

Solution: Let and Given


[ since and are independent ]

[ since and are independent ]

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.

14. Find the correlation co-efficient for the following dataX 10 14 18 22 26 30Y 18 12 24 6 30 36

Solution: Here

.

15. From the following data, find (1) the two regression equations (2) the co-efficient of correlation between the marks in Mathematics and Statistics. (3) the most likely marks in Statistics when marks in Mathematics are 30.

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10 18 180 100 32414 12 168 196 14418 24 432 324 57622 6 132 484 3626 30 780 676 90030 36 1080 900 1296

120 126 2772 2680 3276

Marks in Mathematics

25 28 35 32 31 36 29 38 34 32

Marks in Statistics

43 46 49 41 36 32 31 30 33 39

Solution: Here

25 43 1075 625 184928 46 1288 784 211635 49 1715 1225 240132 41 1312 1024 168131 36 1116 961 129636 32 1152 1296 102429 31 899 841 96138 30 1140 1444 90034 33 1122 1156 108932 39 1248 1024 1521

320 380 12067 10380 14838

The line of regression of on is

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The line of regression of on is

When marks in Mathematics are (ie) when , we have Therefore marks in Statistics =

16. If and are the two regression lines, find the mean values of and . Find the correlation co-efficient between and . Find an estimate of when . Solution: Given -----------------(1) -----------------(2) Since both the lines of regression passes through the mean values and , the point must satisfy the two given regression lines.

-------------(3) -------------(4) Subtracting the equations (3) and (4), we have

Therefore mean values are and .

Let us suppose that equation (1) is the line of regression of on and equation

(2) is the equation of the line of regression of on , we have

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Since both the regression co-efficients are positive, must be positive. Correlation co-efficient = Substituting in (2), we have

.

17. If the joint pdf of is given by , find the pdf of . Solution: Given Consider the auxiliary random variable

Therefore, the joint density function of is given by

The pdf of is given by

.

18. If the joint pdf of is given by , find the pdf

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of .

Solution: Given Introduce the auxiliary random variable

Therefore, the joint density function of is given by

The pdf of is given by

.

19. State and prove central limit theorem. Solution: Statement:

If be a sequence of independent identically distributed random variables with and , and if , then under certain general conditions, follows a normal distribution with mean and variance as n tends to infinity.

Proof: To prove that as , follows the standard

normal distribution, we use uniqueness theorem. “ If two random variables have same probability distribution, then their moment generating function are identical”.

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We know that moment generating function of

. If we prove moment generating function of is , then

follows standard normal distribution with mean and variance . Moment generating function of is,

Because since are independent and identically distributed.

Consider

Therefore,

We know that

Comparing, we have

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Therefore follows standard normal distribution with mean and variance and hence follows normal distribution with mean and variance .

20. The lifetime of a certain brand of an electric bulb may be considered a RV with mean and standard deviation . Find the probability using central limit theorem, that the average lifetime of bulbs exceeds

. Solution: Let represent the life of the bulb. Given and Let denote the mean lifetime of bulbs. By central limit theorem, we have

.

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Unit ii rpq

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