Unformatted text preview: 180A HW 6 Solutions
November 30, 2006 1) Suppose X has the exponential distribution with parameter and Y has the exponential distribution with parameter . Assume X and Y are independent. Let Z = X/(Y + 1). (a) Compute E[Z]. (b) Compute the PDF of Z. Answer (a) Since X and Y are independent, X and 1/(Y+1) are independent, and we have E[Z] = E[X/(Y + 1)] = E[X]E[1/(Y + 1)] = (1/)(
0 1 ey dy). y+1 Note that E[1/(Y + 1)] = 0 (1/y + 1)ey dy does not have a closed form solution. (b) We will find the PDF of Z by finding the CDF of Z, G(z), and then differentiating it: z(y+1) G(z) = P (Z z) = P (X/(Y + 1) z) = P (X z(y + 1)) =
0 0 f (x, y)dxdy Where f(x,y) is the joint density of X and Y . Since X and Y are independent, f (x, y) = fX (x)fY (y) = exy . Thus, z(y+1) G(z) =
0 0 0 (ex  y)dxdy = (ez(y+1)y + ey )dy = ez(y+1)y ey  ] = z 0 ez 1 z [ Thus the PDF of Z, g(z), is: g(z) = G (z) = ez  zez z 2 1 2) Let X and Y be random variables with PDF f (x, y) = x + y, x, y (0, 1), and f (x, y) = 0, otherwise Answer
1 1 1 1 0 E[X] =
0 0 xf (x, y)dxdy =
0 1 0 (x2 + xy)dxdy = (1/3 + y/2)dy = 7/12. E[X 2 ] =
0 1 0 1 x2 (x + y)dxdy =
0 1 0 1 0 1 (x3 + x2 y)dxdy = (1/4 + 1/3y)dy = 5/12. V ar[X] = E[X 2 ]  E[X]2 = 5/12  (7/12)2 = 11/144.
1 1 1 1 0 1 0 E[XY ] =
0 0 xy(x + y)dxdy =
0 (yx2 + xy 2 )dxdy = (y/3 + y 2 /2)dy = 1/3. Cov(X, Y ) = E[XY ]  E[X]E[Y ] = 1/3  (7/12)(7/12) = 1/144. Since our density is symmetric in X and Y , E[Y ] = E[X] = 7/12 and V ar[Y ] = V ar[X] = 11/144. Thus E[Z] = E[X] + E[Y ] = 7/12 + 7/12 = 7/6 and V ar[Z] = V ar[X + Y ] = V ar[X] + V ar[Y ] + 2Cov[X, Y ] = 11/144 + 11/144  2/144 = 20/144 = 5/36. 3) Suppose X1 , X2 are independent, each with the uniform distribution on (0, 1). Compute the probability density function of Y = X1 + X2 . Answer The convolution of X and Y gives: fZ (t) =
 fX (x)fY (t  x)dx. Since 0 fX (x) 1 and 1  t fY (t  x) t for X, Y uniform on [0, 1], we have:
t fZ (t) =
0 1 dx for 0 t 1, dx for 1 t 2,
t1 fZ (t) = fZ (t) = 0 for t > 2 or t < 0. 2 Thus 0 t fZ (t) = 2t 0 t0 0t1 1<t2 t>2 4) Show by induction (on n) that Y = X1 + ... + Xn has the negative binomial distribution with parameters (n, p) when the Xi s are independent, each with the geometric distribution with parameter p. Answer For the base case n=1, note that N egBin(1, p) Geom(p). Now assume Z = X1 +...+Xn 1 N egBin(n  1, p), and let Xn Geom(p). We want to show Z + Xn N egBin(n, p). By the convolution formula, P (Y = k) = P (Z + Xn = k) = all x
t1 x=n1 P (Z = x)P (Xn = k  x) = x  1 n1 p (1  p)x(n1) p(1  p)kx1 = n2
k1 pn (1  p)kn
x=n1 x1 n2 From the hint, letting y = x  1 we get
k1 x=n1 x1 n2 k2 =
y=n2 y n2 = k1 n1 So P (Y = k) = Thus Y N egBin(n, p). 5) Let X1 , ..., Xn be independent, with Xi having the exponential distribution with parameter i . Define Y = n Xi . Compute E(Y k ) for k = 1, ..., 4. i=1 Answer First we will calculate E[Xin ], n = 1, 2, 3, 4: E[X n ] =
0 k1 n p (1  p)kn n1 xn ex dx Using intregation by parts, we get: 3 E[X n ] = ([ xn ex ]0 ) + n 0 nxn1 x e dx = xn1 ex dx = n E[xn1 ] 0 Since E[X] = 1/, 2 E[X] = 3 E[X 3 ] = E[X 2 ] = 4 4 E[X ] = E[X 3 ] = E[X 2 ] = Also, observe that Y 2 is:
n 2 2 6 3 24 4 Y = (X1 + ... + Xn ) =
i=1 2 2 Xi2 +
i=j Xi Xj Y 3 is:
n Y = (X1 + ... + Xn ) = (
i=1 n 3 3 Xi2 +
i=j Xi Xj )(X1 + ... + Xn ) = Xi Xj )(X1 + ... + Xn )) = ((
i=1 n Xi2 )(X1 + ... + Xn )) + ((
i=j 2 Xi Xj ) + ( i=j i=j n (
i=1 Xi3 + Xi2 Xj +
i=j 2 Xi Xj + i=j=k 2 Xi Xj + i=j Xi Xj Xk ) = Xi Xj Xk
i=j=k Xi3 + 3
i=1 and Y 4 is: 4 n Y 4 = (X1 + ... + Xn )4 = (
i=1 n Xi3 + 3
i=j 2 Xi Xj + i=j=k Xi Xj Xk )(X1 + ... + Xn ) = Xi Xj Xk )(X1 + ... + Xn ) =
3 Xi Xj ) + (
i=1 n Xi3 )(X1 + ... + Xn ) + (3
i=j 3 Xi Xj ) + (3 i=j i=j 2 Xi Xj )(X1 + ... + Xn ) + ( i=j=k 2 Xi Xj Xk + 3 i=j=k i=j (
i Xi4 + 2 Xi2 Xj + 3 Xi Xj Xk Xl +
i=j=k=l Xi2 Xj Xk
i=j=k n +
i=j=k 2 Xi Xj Xk +
i=j=k 2 Xi Xj Xk = Xi4 + 4
i i=j=k 3 Xi Xj + 3 i=j 2 Xi2 Xj + 6 i=j=k 2 Xi Xj Xk + i=j=k=l Xi Xj Xk Xl Using these facts, along with the fact that expectation is a linear function and the Xi s are independent, we get E[Y ] is:
n n n E[Y ] = E[
1 Xi ] =
1 E[Xi ] =
1 1/i E[Y 2 ] is:
n n E[Y ] = E[
i=1 2 Xi2 +
i=j Xi Xj ] =
i=1 2/2 + i
i=j 1/(i j ) E[Y 3 ] is:
n E[Y ] = E[
i=1 n 3 Xi3 + 3
i=j 2 Xi Xj + i=j=k Xi Xj Xk ] = 1/(i j k )
i=j=k 3/3 + 3 i
i=1 i=j 2/(i /2 ) + j and E[Y 4 ] is:
n E[Y 4 ] =
i 4/4 + 4 i
i=j=k 3/(i 3 ) + 3 j
i=j 4/(2 2 ) + 6 i j
i=j=k 2/(i j 2 ) + k
i=j=k=l 1/(i j k l ) 5 ...
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 Fall '08
 Castro
 Normal Distribution, Probability, Probability theory, Exponential distribution, Xi Xj Xk, Xi Xj

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