Unit-3

Bivariate Distributions

Q1) Let be two independent N (0, 1) random variables. Define

Where is a real number in (-1, 1).

1. Show that X and Y are bivariate normal.
2. Find the joint PDF of X and Y.
3. Find (X,Y)

S1)

First note that since are normal and independent they are jointly normal with the joint PDF

1. We need to show aX + bY is normal for all. We have

Which is the linear combination of and thus it is normal.

b.     We can use the method of transformations (theorem 5.1) to find the joint PDF of X and Y. The inverse transformation is given by

We have Where, Thus we conclude that To find FIRST NOTE Therefore,

Q2) Let X and Y be jointly normal random variables with parameters Find the conditional distribution of Y given X =x.

S2) One way to solve this problem is by using the joint PDF formula since X N we can use

Thus given X=x, we have And, Since are independent, knowing does not provide any information on . We have shown that given X=x,Y is a linear function of , thus it is normal. Ln particular We conclude that given X=x,Y is normally distributed with mean and variance

Q3) A die is tossed thrice. A success is getting 1 or 6 on a toss. Find the mean and variance of the number of successes.

S3)

Probability of success probability of failures

Probability of no success= probability of all three failures

Probability of one successes and two failures

Probability of Two successes and one failure

Probability of three successes

	1	2	3
	4/9	2/9	1/27

Mean

Variance,

Q4) The probability density function of a variate X is

S4) (I) If X is random variable then

hus minimum value of k =1/30

Q5) Random variable X has the following probability function

x	0	1	2	3	4	5	6	7
P (x)	0	k	2k	2k	3k

(i)                Find the value of the k

(ii)             Evaluate P (X < 6), P (X≥6)

(iii)         

S5) (i) if X is a random variable then

(ii)P (X < 6) =P( X=0) +P(X=1)+P(X=2)+ P(X=3) +P(X=4) + P (X=5)

(iv)           

Q6) A bag contains 12 pens of which 4 are defective. Three pens are picked at random from the bag one after the other.

Then find the probability that all three are non-defective.

S6) here the probability of the first which will be non-defective = 8/12

By the multiplication theorem of probability,

If we draw pens one after the other then the required probability will be-

Q7) An urn contains 3 white and 4 red balls and an urn lI contains 5 white and 6 red balls. One ball is drawn at random from one ofthe urns and isfound to be white. Find the probability that it was drawn from urn 1.

S7) Let : the ball is drawn from urn I

: the ball is drawn from urn II

: the ball is white.

We have to find

By Bayes Theorem

... (1)

Since two urns are equally likely to be selected, (a white ball is drawn from urn )

(a white ball is drawn from urn II)

From(1),

Q8) Three urns contains 6 red, 4 black, 4 red, 6 black; 5 red, 5 black balls respectively. One of the urns is selected at random and a ball is drawn from it. Lf the ball drawn is red find the probability that it is drawn from the first turn.

S8)

Let : the ball is drawn from urn 1.

: the ball is drawn from urn lI.

: the ball is drawn from urn 111.

: the ball is red.

We have to find .

By Baye’s Theorem, ... (1) Since the three urns are equally likely to be selected Also (a red ball is drawn from urn ) (R/ ) (a red ball is drawn from urn II) (a red ball is drawn from urn III) From (1), we have

Q9) ln a bolt factory machines and manufacturerespectively 25%, 35% and 40% of the total. Lf their output 5, 4 and 2 per cent are defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by machine B.?

S9) bolt is manufactured by machine

: bolt is manufactured by machine

: bolt is manufactured by machine

The probability of drawing a defective bolt manufactured by machine is (D/A)

Similarly, (D/B) and (D/C)

By Bayes’ theorem