undefined | unit 8 concept of joint probability

EM-IV

Unit-8Concept of joint probabilityQ1) Find the joint distribution of X andY, which are independent random variables with thefollowing respective distributions:

	1	2
P [ X =	0.7	0.3

And

	-2	5	8
P [ Y =	0.3	0.5	0.2

A1)

Since X and Y are independent random variables,

Thus the entries of the joint distribution are the products

of the marginal entries

Q2) The following table represents the joint probability distribution ofthe discrete random variable (X, Y):

X/Y	1	2
1	0.1	0.2
2	0.1	0.3
3	0.2	0.1

Then find-i) The marginal distributions.ii) The conditional distribution of X given Y = 1.iii) P[(X + Y) < 4].A2)i) The marginal distributions.

X/Y	1	2	p(x) [totals]
1	0.1	0.2	0.3
2	0.1	0.3	0.4
3	0.2	0.1	0.3
p(y) [ totals]	0.4	0.6	1

The marginal probability distribution of X is-

X	1	2	3
p(x)	0.3	0.4	0.3

The marginal probability distribution of Y is

Y	1	2
p(x)	0.4	0.6

ii) The conditional distribution of X given Y = 1.

The conditional distribution of X given Y = 1 is-

X	1	2	3
	¼	1/4	½

(iii) The values of (X, Y) which satisfy X + Y < 4 are (1, 1), (1, 2) and (2, 1)only.

Which gives-

Q3) In a lottery, m tickets are drawn at a time out of a tickets numbered from 1 to n. Find the expected value of the sum of the numbers on the tickets drawn.A3)

Let be the variables representing the numbers on the first, second,…nth ticket. The probability of drawing a ticket out of n ticket spelling in each case 1/n, we have