8. For each i = 1, 2, ..., 10, Xi is a random variable that gives 0 or 1 if the ith toss of a fair coin came up T or H, respectively. Let X = X1 + X2 + ... + X 10. a. Compute the expected value E(X) and variance V(X) of X. (5) b. What is the probability function of X? [10]
8. For each i = 1, 2, ..., 10, Xi is a random variable that gives 0 or 1 if the ith toss of a fair coin came up T or H, respectively. Let X = X1 + X2 + ... + X10 a. Compute the expected value E(X) and variance V(X) of X. [5] b. What is the probability function of X? (10)
7. A jar contains 6 white beads and 3 black beads. Beads are chosen randomly from the jar one at a time until the third time a black bead turns up. a. Suppose that each bead is replaced before the next is chosen. How many beads should you expect to be chosen in the course of the experiment? [5] b. Suppose that if a bead is white, it is not replaced before the next bead is chosen, but if it...
2. (8 Marks - 2 points each) Suppose you toss a fair coin four times. Let the random variable X be the number of tails (T) obtained. Also, let E and V denote the mean and variance respectively. a) Compute E(X) and V(X). b) Compute E(5X - 6) and V (6 +5X).
Let xi, i 1, 2, 3, , be a sequence of nonnegative numbers such that Σ x.-1 and consider the random variable X whose probability function is defined by: x, for x=x1, x2, X3, 0, for all other x What is the variance of X? i= 1
Let X be an exponential random variable with parameter A > 0, and let Y be a discrete random variable that takes the values 1 and -1 according to the result of a toss of a fair coin Compute the CDF and the PDF of Z = XY Let X be an exponential random variable with parameter A > 0, and let Y be a discrete random variable that takes the values 1 and -1 according to the result of...
2. Let S be the sample space of a single toss of a fair coin. Define the sequence of random variables X, on S as follows: (I Ifs-T (a) Are X1.x2 . Convergent almost surely? (b) Find P((s E S : limx,(s)-1)). 2. Let S be the sample space of a single toss of a fair coin. Define the sequence of random variables X, on S as follows: (I Ifs-T (a) Are X1.x2 . Convergent almost surely? (b) Find P((s...
1. (6 pts) Consider a non-negative, discrete random variable X with codomain {0, 1, 2, 3, 4, 5, 6} and the following incomplete cumulative distribution function (c.d.f.): 0 0.1 1 0.2 2 ? 3 0.2 4 0.5 5 0.7 6 ? F(x) (a) Find the two missing values in the above table. (b) Let Y = (X2 + X)/2 be a new random variable defined in terms of X. Is Y a discrete or continuous random variable? Provide the probability...
Let X1, X2, and X3 be uncorrelated random variables, each with 4. (10 points) Let Xi, X2, and X3 be uncorrelated random variables, each with mean u and variance o2. Find, in terms of u and o2 a) Cov(X+ 2X2, X7t 3X;) b) Cov(Xrt X2, Xi- X2)
10. Let X be a continuous random variable with probability density function -2 хе x > 0 h(z) = { { 0 x < 0 a. Verify that h(x) is a valid probability density function. [7] b. Compute the expected value E(X) and variance V(X) of X. [8]