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Probability theory
Mid Semester Examination
KV
(1) Let η be a random variable with distribution function F (). Describe
the distribution of the random variable F (η).
4 marks
(2) You throw k balls into n bins. If k > n, what is the probability that each
bin has atleast one ball?
5 marks
(3) Let X be a random variable with normal distribution N (0, 1). What is
the density of the random variable Y = eX ?
5 marks
(4) Let f (x, y) be the joint density of a pair of random variables X, Y , given
by f (x, y) = xe−x(y+1) , 0 ≤ x, y ≤ ∞.
(a) Verify that f is a joint distribution
(b) Find the marginal densities of X, Y .
(c) Compute fX|Y (x|y) and fY |X (y|x)
(d) Show that the distribution function F (a) of the random variable
XY is given by 1 − e−a .
8 marks
(5) You have a coin which has a certain probability p ≥ a, of heads showing up. In order to estimate p you toss a coin N times, see how many
times you get heads and divide by N to get a quantity p̄. How large
should N be so that your estimate p̄ satisfies P r[|p − p̄| > p] < δ for
any choice of 0 < a, , δ. Use Chernoff bounds.
6 marks
(6) Let X1 , x2 , . . . be independent random variables, each with a uniform
distribution in [0,1].
(a) Compute P r[X1 + X2 . . . Xn ≤ x] for 0 ≤ x ≤ 1.(Hint:Use induction on n)
1
(b) Let Y be the random variable which equals the smallest number
k for which X1 + X2 . . . Xk exceeds 1. Compute the distribution
function of Y and E(Y ).
6 marks
(7) A factory has produced n robots, each of which is faulty with probability p. To each robot a test is applied which detects the fault, if
present, with probability q. Let X be the number of faulty robots
and Y , the number detected as faulty. Assuming that robots are independently faulty and that the tests are also independent, show that
.
6 marks
E(X|Y ) = np(1−q)+(1−p)Y
1−pq
2