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Helpful Formulas ˆ Gamma-function is de ned for any a > 0 as follows: 􀀀(a) = R1 0 ya􀀀1  e􀀀y dy ˆ For any a > 0; the identity 􀀀(a + 1) = a  􀀀(a) holds ˆ If a  1 is a whole number, then 􀀀(a) = (a 􀀀 1)! ˆ Random variable Y has gamma-distribution Gamma[a; ] with shape a > 0 and scale  > 0 if its density is f(y) = fY (y) = 1 a  􀀀(a)  ya􀀀1  e􀀀y= for y > 0 and f(y) = 0 elsewhere: ˆ First two moments of gamma-distributed random variable Y  Gamma[a; ] are: E[Y ] = a   and E[Y 2] = a(a + 1)  2 =) Var [Y ] = a  2 ˆ For any a > 0 and k > 0; Z 1 0 ya  e􀀀ky dy = 􀀀(a + 1) ka+1 ˆ Beta-function is de ned for a > 0 and b > 0 as follows: B(a; b) = R 1 0 ua􀀀1  (1 􀀀 u)b􀀀1 du ˆ Beta-function satis es the identity: B(a; b) = 􀀀(a)  􀀀(b) 􀀀(a + b) ˆ Random variable U has Beta [a; b] distribution means that its density is f(u) = fU(u) = 1 B(a; b)  ua􀀀1(1 􀀀 u)b􀀀1 = 􀀀(a + b) 􀀀(a)  􀀀(b)  ua􀀀1  (1 􀀀 u)b􀀀1 for 0 < u < 1 and fU(u) = 0; elsewhere. ˆ First two moments of beta-distributed random variable U  Beta [a; b] are: E[U] = a a + b and E[U2] = a(a + 1) (a + b)(a + b + 1) =) Var [U] = a  b (a + b)2(a + b + 1) 2 Problem 6 [10 points = 5 + 5] A continuous random variable T has density function fT (t) = f(t) = 9t  e􀀀3t for t > 0 and f(t) = 0; elsewhere: Conditionally, given T = t; a random variable W is uniformly distributed over the interval (0; t); so that (WjT = t)  Unif (0; t) Derive marginal density and cumulative distribution function for W: ˆ fW(w) = ˆ FW(w) = Solution 8 Problem 7 [10 points = 5 + 5] A discrete random variable N has probability mass function de ned for integer n  0 as follows: fN(n) = P[N = n] = 6n n!  e􀀀6 Given N = n; a discrete random variable K has conditional probability mass function de ned for integer values (0  k  n) as fKjN(kjn) = P[K = kjN = n] = n! k!(n 􀀀 k)!  2k 3n Derive marginal probability functions for K and M = N 􀀀 K ˆ P[K = k] = ˆ P[N 􀀀 K = m] = Solution 9 Problem 8 [10 points = 5 + 5] Assume that independent Poisson distributed variables, M and K have probability mass functions de ned as fM(m) = P[M = m] = 2m m!  e􀀀2 for m  0 and fK(k) = P[K = k] = 3k k!  e􀀀3 for k  0: ˆ Find conditional probability function for (MjM + K = 10) P[M = mjM + K = 10] = ˆ Evaluate conditional covariance between M and K; given that M + K = 10 Cov [(M;K)jM + K = 10] = Solution 10 Problem 9 [10 points = 5 + 5] A continuous random variable Y has density function de ned for y > 0 as follows: f(y) = fY (y) = 216  y3  e􀀀6y and f(y) = 0; elsewhere. Given Y = y; a discrete random variable N has conditional probability mass function de ned for integer n  0 as follows: fNjY (njy) = P[(N = n)jY = y] = yn n!  e􀀀y and fNjY (njy) = 0; otherwise. Evaluate marginal expectation and variance for N ˆ E[N] = ˆ Var [N] = Solution 11 Problem 10 [10 points = 5 + 5] A continuous random variable Q has density function de ned for 0 < q < 1 as follows: f(q) = fQ(q) = 60  q3  (1 􀀀 q)2 and f(q) = 0; elsewhere. Given Q = q; a binary random variable U has conditional probability mass function P[U = 1jQ = q] = fUjQ(1jq) = q and P[U = 0jQ = q] = fUjQ(0jq) = 1 􀀀 q ˆ Evaluate marginal distribution for U; P[U = 1] = P[U = 0] = ˆ Derive conditional densities fQjU(qj1) = fQjU(qj0) = Solution 12

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