2 A discrete random variable Y takes the values -1, 0 and 1 with probabilities 10, 1-0 and 10 respectively. Let Y₁ and Y₂ be two independent random variables, each with the same distribution as Y. (a) List the possible values of {Y₁, Y₂) that may arise and calculate the probability of each. Verify that your probabilities sum to unity. (b) By calculating the value of (Y₂ - Y₁)² for each possible pair (Y₁, Y₂), determine the sampling distribution of (Y₂ - Y₁)². ... (c) Show that X = (Y2Y₁)² is an unbiased estimator for 0, and find Var(X) as a function - of 0. (d) Now suppose that Y1, Y2..... Y, are n independent observations on the random variable Y. Since 0 is the probability that Y is not zero, a possible estimator for 0 is the proportion, Z, of non-zero values among Y₁, Y2,..., Y. Write down the mean and variance of Z, and state, giving your reasons, which of X and Z you would prefer as an estimator for 8 when n = 2.

Let X and Y be independent discrete random variables and suppose that X+Y=2. Show that X and Y are constant random variables.

Coin A has a probability of head equal to 1/4 and probability of tail equal to 3/4 and coin B is a fair coin. Each coin is flipped four times. Let the random variable X denote the number of heads resulting from coin A and Y denote the resulting number of heads from coin B. (a) What is the probability that X = Y = 2 ? (b) What is the probability that X = Y ? (c) What is the probability that X > Y? (d) What is the probability that X + Y< 5?

An ordinary (fair) coin is tossed 3 times. Outcomes are thus triples of "heads" (h) and "tails" (t) which we write hth, ttt, etc. For each outcome, let N be the random variable counting the number of heads in each outcome. For example, if the outcome is hhh, then N (hhh) = = 3. Suppose that the random variable X is defined in terms of N as follows: X=6N-2N²-3. The values of X are given in the table below. Outcome hhh hth hht thh htt tth ttt tht Value of X-3 1 1 1 1 1 -3 1 Calculate the probabilities P (X=x) of the probability distribution of X. First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row. Value X of X P(X=x) 0 0 0 00 X

Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(X)=-1 E(Y) = 4 E (Z)=0 Var(X)=20 Var(Y) = 27 Var (Z) = 38 Compute the values of the expressions below. E (5Z-2) = 0 2Z+X -5 Var(2X-3) = 0 E = E (32²) = 0 Explanation Check X O hp Ⓒ2022 McGraw Hill LLC. All Rights Reserved. Term

Let Y be a binomial random variable with n = 10 and p = 0.3. (a) P(3 < Y < 5) = P(3 ≤ Y < 5) = (b) P(3 < Y ≤ 5) = P(3 ≤ Y ≤ 5) =

A particle executes an unrestricted simple random walk on the line with p=0.7 and q=0.3. Its position at time n is represented by the random variable Xn, n = 0, 1,...; Xo = 0. (a) Calculate the values of the following probabilities. (i) P(X5 = -2) (ii) P(X6 = 2) (b) Calculate the probability that the particle returns to its starting point for the first time after 8 steps. (c) After the particle has been moving for some time, it is observed to be located at the point +3. (i) Calculate the probability that the particle will ever return to the origin from the point +3. (ii) Calculate the probability that starting from the point +3, the particle will visit the point +8 without first visiting the origin.

Let random variable X be the number of users communicating with a cellular base station within a given time interval. Its PMF is :

The random variable X takes values -1, 0, 1 with probabilities 1/8, 3/8, 4/8 respectively. a) Write the CDF of X. b) Write the PMF of Y = X² + 2. %3D c) Compute E(Y).

A shipment of 7 television sets contains 2 defective sets. A hotel makes a random purchase of 3 of the sets. If X is the number of defective sets purchased by the hotel, (a) find the probability distribution of X, express it in a table format. Probability distribution of X X P(X) (b) Find the cdf of X. (c) Find the E(X) and Var(X)