(2) Consider rolling three dice. Let X1, X2, and Xj be the values which appear on the three dice, respectively. Let Y b...
(4) Consider rolling three dice. Let X1, X2, and X3 the values which appear on the three dice, respectively. Let Y be the maximum out of all three dice. (a) Find the conditional PMF py)xi (yr) (b) Find the probability that the maximum of all three dice is 4 given that the first die is a 3. (c) Find the probability that the maximum of all three dice is 3 given that the first die is a 3 (4) Consider...
(4) Consider rolling three dice. Let X1, X2, and X3 the values which appear on the three dice, respectively. Let Y be the maximum out of all three dice. (a) Find the conditional PMF py)xi (yr) (b) Find the probability that the maximum of all three dice is 4 given that the first die is a 3. (c) Find the probability that the maximum of all three dice is 3 given that the first die is a 3 (4) Consider...
O. Let X1 and X2 be two random variables, and let Y = (X1 + X2)2. Suppose that E[Y ] = 25 and that the variance of X1 and X2 are 9 and 16, respectively. O. Let Xi and X2 be two random variables, and let Y = (X1 X2)2. Suppose that and that the variance of X1 and X2 are 9 and 16, respectively E[Y] = 25 (63) Suppose that both X\ and X2 have mean zero. Then the...
Given random variables X1, X2, Y with E[Y | X1, X2] = 5X1 + X1X2 and E[Y 2 | X1, X2] = 25X2 1X2 2 + 15, find E[(X1Y + X2) 2 | X1, X2]. ㄨ竺Bin(2.1/4). Suppose X and Y are independent random variables. Find the expected value of YX. Hnt: Consider conditioning on the events (X-j)oj0,1,2. 9. Given random variables XI,X2, Y with E'Y | XiN;|-5X1 + X1X2 and Ep2 1 X1,X2] 25XX15, find 10. Let X and Y...
2.Let Xj,X,, Xj, X4, Xj be a random sample of size n-5 from a Poisson distribution with mean ?. Consider the test Ho : ?-2.6 vs. H 1 : ? < 2.6. a)Find the best rejection region with the significance level a closest to 0.10 b) Find the power of the test from part (a) at ?= 2.0 and at ?=1.4. c) Suppose x1-1, x2-2, x3 -0, x4-1, x5-2. Find the p-value of the test.
Consider the procedure of rolling a pair of dice 6 times and let x be the random variable consisting of the number of times the sum of the results is 7. The following table describes the probability distribution of x. X P(X) 0 0.334898 1 ¿? 2 ¿? 3 0.053584 4 0.008038 5 0.000643 6 0.000021 a) Find the missing probabilities b) It would be unusual to roll a pair of dice six times and get at least three times...
We roll two dice. Assume all 36 possibilities are equally likely. Let X1 and X2 be the result of the first and second die, respectively. Let S be the sum of the scores, that is S = X1 + X2. Calculate the following: (a) P(S = k), for k 2,3,... 12. (b) P(X1 = 2 S = k) for k = 2,3,... 12. (c) P(X1 = 6|S = k) for k = 2, 3, ... 12.
3) Let (x, y), (X2, y2), and (X3. Y3) be three points in R2 with X1 < x2 < X3. Suppose that y = ax + by + c is a parabola passing through the three points (x1, yı), (x2, y), and (x3, Y3). We have that a, b, and c must satisfy i = ax + bx + C V2 = ax + bx2 + c y3 = ax} + bx3 + c Let D = x X2 1....
Let X1 be a random variable whose value is the result of rolling an 8- sided die, and X2 a random variable whose value is the result of rolling a 12-sided die. (1) Find E(X1 + X2). (2) Find E(X{ + Xž). (3) Find V(X1 + X2).
2. Let X1 and X2 be the numbers showing when two fair dice are thrown. Define new random variables X = Xi-X2 and Y = X1 + X2. Show that X and Y are uncorrelated but not independent. Hint: To show lack of independence, it is enough to show that PlX = j, Y = k]メPIX = j] . PY = k] for one pair (j, k); try the pair (0.2).]