Each of the random variables X and Y takes only 3 values {1,2,3} with the following probabilities: 1 1 0 1/6 1/6 y 2 1/6 0 1/6 3 1/6 1/6 0 2.
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- Q10. The probability model for a random variable A is [% ,a=-1 P₁(a)=% a=1 0, otherwise The conditional probability model for random variable B is: %,b=0 P (b/-1)=% .b=1 0, otherwise P (b/1)=,b=1 0, otherwise a) What is the probability model for random variables A and B If 4-1, what is the conditional expected value E[BA-11? c) If B-1, what is the conditional PMF Pa (at)? b) d) If B-1, what is the conditional variance Var [4| B=1]? 110 markeAn 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 hht, then N (hht) =2. Suppose that the random variable X is defined in terms of N as follows: X=6N-2N²-4. The values of X are given in the table below. Outcome hth ttt tth tht hhh thh htt hht Value of X 0 -4 0 0 -4 0 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 00In a XYZ car company car saleswoman has to sell 1 car. She is provided with a very large (infinite for practical purposes) list of customers. She approaches customers sequentially according to the list. The probability that she makes a successful sale to any given customer is 0.2. She stops as soon as she sells the car. Suppose that all the customers behave independently of each other. Find the expected value and variance of the total number of customers the saleswoman has to approach. Select one: a. EIX) = 20 and V(X) = 5 b. None of these O c. E(X) = 4 and V(X) = 20 d. EIX) = 5 and V(X) = 10 O e. E(X) = 5 and V(X) = 20 The correct answer is: E(X) = 5 and V(X) = 20
- X and Y are two random variables and k is a constant. If E(kX + Y) = 16, E(X + Y) = 6 and E(X + kY) = 8, find the possible value of k. Hence, find E(X) and E(Y).3) Let us assume that we have a five sided die. The sides of the die are marked with the numbers -2, –1,0, 1,2 and the probabilities are as follows: P(X = -2) = 0.05, P(X = -1) = 0.2, P(X = 0) = 0.3, P(X = 1) = 0.4, P(X = 2) = 0.05 a) Calculate E[X²]. b) Calculate VAR[X²j. c) Calculate approximately the following probability: P(X{ + X3 + ...+Xf00 < 111), where X1, X2,... are i.i.d. variables all with the same probability as X.Q6a
- If x is functionally related to A and B as x = A + B, and if A and B are random variables, then x is also a random variable. True or FalseIf x and y are two independent random variables, show that +y:x-y={r?:x+}-3:x+, where rs+y:x-y denotes the coefficient of correlation between x + y and x – y. Ty:x+ Tx+y:x-yIf X is a continuous random variable that takes on values between 10 and 40, then the P(X = 15.5) = 0 True or false
- 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 R be the random variable counting the number of heads in each outcome. For example, if the outcome is hhh, then R (hhh)=3. Suppose that the random variable X is defined in terms of R as follows: X= 2R-2R^2-3. The values of X are given in the table below.Let X be a random variable such that E(X – 2)² = 10 and E(X+1)² = 4, then а. Е(X) 1 = - b. Е(X) 3D с. Е(X) %3D 4 d. E(X)= -For two random variables X and Y, JXY (*, y) = 0.15 8 (x + 1) · 8 (v) -+ 0,1 8 (x) ·8 (v) + 0.1 8 (x) · ò (y – 4) + 0.4 8 (x - 1) - 8 (y + 2) + 0.2 8 (x - 1) 8 (y - 1) + 0.05 8 (x - 1) 8(y-3) Find the correlation