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M3

Unit-3Bivariate Distributions Q1) Let be two independent N (0, 1) random variables. DefineWhere is a real number in (-1, 1).
  • Show that X and Y are bivariate normal.
  • Find the joint PDF of X and Y.
  • Find (X,Y)
  • S1)First note that since are normal and independent they are jointly normal with the joint PDF
  • 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

    1. 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 failuresProbability of no success= probability of all three failuresProbability of one successes and two failures Probability of Two successes and one failureProbability of three successes

    1

    2

    3

    4/9

    2/9

    1/27

     Mean Variance, Q4) The probability density function of a variate X is

    X

    0

    1

    2

    3

    4

    5

    6

    P (X)

    k

    3k

    5k

    7k

    9k

    11k

    13k

     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 = 8/12By 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