Annual number of tornadoes in countyQ 2 Annual number 0 0.12 0.06 0.05 0.02 of t

Question

Annual number of tornadoes in countyQ 2 Annual number 0 0.12 0.06 0.05 0.02 of tornadoes 0.13 0.15 0.12 0.03 in county P 2 0.05 0.15 0.10 0.02 (a) Find the conditional probability distribution of Q, given that P-o (b) Find E(QP=0). (c) Find V(QP 0). (d) Find the conditional probability distribution of Q, given that P-1. (e) Find E(QIP 1). (f) Find V(Q[P=1). (g) Find the conditional probability distribution of Q, given that P = 2. (h) Find E(QIP 2). (i) Find V(QIP 2). (j) Find E(Q) using E(Q) = E[E(QIP). (Hint: This is the same as E(Q)-Ep P(P-pE(QIP = p).) (k) Find E() using the marginal distribution of Q. the marginal distribution of Q 7. (1 point bonus) Find V(O) using 8. (4 points bonus) Find V( using V(Q) EIV(QIP)]+VIE(QIP)). (To match the answer to question 7 you will ne to keep as many decimal places as possible.) First we find E[V(QIP)] as Now we find VIE(QIP)) as ELLE(QIP)))-EEQP2 First we find EllE(QIP))2) as Now we find {E|E(QIP)))2 as {E[E(QlP)l = {E(Q))2 =

Explanation / Answer

7.

Marginal distribution of Q is given as,

P(Q=0) = P(P=0, Q=0) + P(P=1, Q=0) + P(P=2, Q=0) + P(P=3, Q=0)

P(Q=0) = 0.12 + 0.13 + 0.05 = 0.3

P(Q=1) = P(P=0, Q=1) + P(P=1, Q=1) + P(P=2, Q=1) + P(P=3, Q=1)

P(Q=1) = 0.06 + 0.15 + 0.15 = 0.36

P(Q=2) = P(P=0, Q=2) + P(P=1, Q=2) + P(P=2, Q=2) + P(P=3, Q=2)

P(Q=2) = 0.05 + 0.12 + 0.10 = 0.27

P(Q=3) = P(P=0, Q=3) + P(P=1, Q=3) + P(P=2, Q=3) + P(P=3, Q=3)

P(Q=3) = 0.02 + 0.03 + 0.02 = 0.07

E(Q) = 0 * P(Q=0) +  1 * P(Q=1) +  2 * P(Q=2) +  3 * P(Q=3)

= 0 * 0.3 + 1 * 0.36 + 2 * 0.27 + 3 * 0.07 = 1.11

E(Q2) = 02 * P(Q=0) +  12 * P(Q=1) +  22 * P(Q=2) +  32 * P(Q=3)

= 0 * 0.3 + 1 * 0.36 + 4 * 0.27 + 9 * 0.07 = 2.07

V(Q) = E(Q2) - [E(Q)]2 = 2.07 - 1.112 = 0.8379

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