Probability Mid-Term Paper-2009
All questions carry equal marks
(1) A laboratory blood test is 95% effective in detecting a certain disease when it is, in fact, present. However, the test also yields a ‘false positive’ result for 1% of the healthy person tested (That is if a healthy is tested, then, with probability 0.01, the test result will imply he has the disease). If 0.5% of the population actually has the disease; what is the probability of a person has a disease given that the test result is positive.
(2) State and prove Baye’s theorem.
(3) Suppose all n men at a party throw their hats in the center of the room. Each men then randomly selects a hat. Show that the probability that none of the n men selects his own hat is[math]\frac{1}{2!}-\frac{1}{3!}--------- \frac{(-1)^n}{n!}[/math]
(4) Consider two boxes, one containing one black and one white marble, the other two black and one white marble, A box is selected at random and a marble is drawn at random from the selected box. What is the probability that the marble is black
.
(5) An urn contains b black balls and r red balls. One of the ball is drawn at random but when it is put back in the urn C additional balls of the same colour are put in with it. Now suppose that we draw another ball. Show that the probability that the first ball drawn was black given that the second ball drawn was red is [math]\frac{b}{b+\frac{b}{c}}[/math]
(6) For a fixed event B, show that the collection P (A\B), defined for all events A, satisfied the three conditions for a probability. Conclude from this that P(A/B)=P(A/BC)P(C/B)+P(A/B{C^C})P({C^C}/B)
Then directly verify the preceding equation.
(7) Give an example where the events are pairwise independent but are dependent jointly.
(8) An urn contains n white balls numbered 1 to n, n black balls numbered 1 to n and n red balls 1 to n. Two balls are drawn at random without replacement. Find the chance that both balls are of the same colour or bear the same number.
(9) Let X and Y are indepednet gamma random variable with parameters (α,λ) and (β,λ) respectively. Compute the joint density of U = X+Y and V = .
(10) Let X1, X2, --, Xn are independent and identically distributed, with expected value μ and variance σ2. then show that cov .
(11) Find the moment generating function of exponential random variable and then reduce the mean and the variance.
(12) Suppose that an airplane engine will fail, when in flight, with probability 1-p independently from engine to engine, suppose that the airplane will make successful flight if at least 50% of its engines remains operative. For what values of p is a four-engine plane preferable to a two-engine plane.
(13) State and prove central- limit theorem.
(14) A point is uniformly distributed within in disc of radius 1. That is, its density is f (x,y) = C, O ≤ x2 + y2 ≤ 1
Find the probability that its distance from the origin is less than X, O ≤ x ≤ 1.
(15) Let the probability density of X is given by
C(4x-2x2), 0< x<2 =
O , Otherwise
(a) What is the value of C ? (b) P (16) The joint distribution of X and Y is f (x, y) = , o < x < y,o < y < ∞. Find the joint and marginal distribution of X and X + Y. (17) Find the moment generating function of gamma distribution and deduce its mean and the variance. (18) Calculate the moment generating function of the uniform distribution on (0,1). Obtain E [X] and Var [X] by differentiating. (19) Let X be a positive random variable having density function f(x). If f(x) ≤ C for all x, show that, for a >o, ≥ 1 - ac.
(20) If X1, X2, --, Xn are independent variates, then Var (a1X1 + a2X2+--+anXn) =
