Question

please explain all rhanks

Search 19:24 If the probability that head is 1/2 and the probability that the back is 1/2, coin is repeatedly throws twice w1 (H,H) w2=(T,H) w3 (H,T) w4= (T,T) The sample space is {w1,w2,w3,w4} The random variable X: R and the random variable Y: R for all we, the probability P is and is defined P ((w)) X (w) 0 for we{w1,w3} X(w)=1, for w e (w2, w4} Y (w) 0, for we{w1,w2} Y(w)1, for w{w3, w4} Further, the random variable Z: R is defined by Z (w) max X (w), Y (w)} with respect to ωΕΩ. 1. Show P (X i, Z k) for i, k 0,1,2 2. Show P (Z k) for k 0,1 3. Find E [X] and E [Z] 4. Find E [XZ] 5. Find E[X | Z 1] 6. Find the variance V of X [V] and the variance VIZ] 7. Find the correlation coefficient p (X, Z) of X and Z

Answer 1

1. P(X 0, Z 0) P(X 0, Y 0) P(w}) 4 1 P(X 0, Z 1)P(X 0, Y 1) P(w3}) 4 P(X 1, Z 1) P(X 1, Y 0)P(X 1, Y = 1) 2 1 P((w2}P(fw) 4 2 1 2. P(Z 0) P(X 0, Z 0) 4 3 P(Z 1) P(X 0, Z 1)P(X 1, Z 1) 4 1 3. E(X) P(X 1) P(fw2, w}) P({w2}) P(fw4}) 2 E(Z) P(Z 1) 4. E(XZ) P(X 1, Z 1) 1, Z 1) P(X 1/2 2 5. E(X|Z 1) P(X 1Z 1) P(Z 1) 3/4 3 1 1 6. V(X) E(X2) E2(X) P(X 1) 4 3 V(Z) E(Z2) E2(Z) 16 E(XZ) E(X) E(Z) V(X)V(Z) Cov(X, Z) X 2 2 7. p(X, Z) _ VV(X)V(Z) 3 X 16 l