[6] Let z be a standard normal random variable. Compute the following probabilities. (a) P(-1.23 s z s 2.58)| (b) P(z > 1.32) (c) P(z 2-1.63)
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- 1. Determine the requested probabilities. -2 -1 1 2 f(x) 0.2 0.4 0.1 0.2 0.1 a) P(xs 2) b) P(x > -2) c) P(-1 sx< 1) d) P(x = 2 or xs-1)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) =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 tails in each outcome. For example, if the outcome is tth, then N (tth)=2. Suppose that the random variable X is defined in terms of N as follows: X=N²-2N-2. The values of X are given in the table below. Outcome ttt htt hhh tht tth hth hht thh Value of X 1 -2 -2 -2 -2 -3 -3 -3 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 0 0 00 X Ś
- Q1 Suppose X and Y are discrete random variables with the following joint p.m.f., where any letters denote probabilities that you might need to figure out. f (x, y) X = -3 X = 0 X = 5 P(Y = y) Y = 1.6 a 0.2 0.1 0.3 Y = 27 P(X = x) 0.3 d 0.2 f e (a) Find P(X >0) (b) Find E(X) and E(Y) (c) Find Var(X) and Var(Y) (d) Find Cov(X, Y) (e) Given that Y= 1.6, find the conditional p.m.f of XLet x be a poisson random variable with mean = 6.5. Find the probabilities of x using the poisson formula. A) P(x=0) B) P(x=1) C) P(x=2) D) P(xLet X be a normal random variable with = 9 and o 1.85 and Y be a normal random variable with u = 3 and o 0.45. Assume X and Y are independent. Find the following probabilities: (a) P(X 8, Y 3.75) (d) P(X < 12, 2.1 ≤Y ≤ 3.8)
- Let X and Y denote two random variables. Which of the following can be used to compute Var(X)? A. E[Var(X|Y)] + Var(Var(X|Y)) B. E[E[X|Y]] + Var(Var(X|Y)) C. E[Var(X|Y)] + Var(E[X|Y]) D. Var(E[X|Y]) + Var(Var(X|Y))Random Variables X and Y are independent. They have probabilities given here: (attached) It is understood that ‘m’ has a numeruc value. Let T be a random variable equal to the sum of X and Y. Find P(T=4).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 tails in each outcome. For example, if the outcome is hth, then N (hth) = 1. Suppose that the random variable X is defined in terms of N as follows: X=2N² − 6N-1. The values of X are given in the table below. Outcome thh tth hhh hth ttt htt hht tht Value of X-5 -5 -1 -5 -1 -5 -5 -5 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 00 X
- If x is a binomial random variable, compute p(x) for each of the following cases: (a) n = 6, x = 2, p = 0.9 p(x) = (b) n = 6, x = 1, p = 0.3Suppose 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(Y) = 4 Var(X) = 24 Var(Y) 18 Var(z)=7 E(X)= -3 E(Z) =-1 Compute the values of the expressions below. E(-4-5X)= 0 %3D ? ()- 0 Var(-3+2.X) = 0Consider a random variable x N(6, 4). Then Prob(X = 12) equals: %3D 0.0 O 0.25 O 0.5 O 0.75 O 0.32