Exercise 5.12. Let n∈N and S={0,1,...,n−1}, and suppose that P({s}) = 1/n for each s∈S. Let...
Exercise 5.12. Let n∈N and S={0,1,...,n−1}, and suppose that P({s}) = 1/n for each s∈S. Let X:S→R be the random variable defined by X(s) =s.
i) Find a closed form formula for MX(z), which does not use sigma notation or any other iterative notation.
(ii) Calculate E[X].
(iii) Calculate Var[X].
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Exercise 5.12. Let n∈N and S={0,1,...,n−1}, and suppose that
P({s}) = 1/n for each s∈S. Let...
CPoisson can not be determined. distribution P(np) ) Suppose X~N(0,1) and YN(24), they are independent, then...
CPoisson can not be determined. distribution P(np) ) Suppose X~N(0,1) and YN(24), they are independent, then (is incorrect. DX+Y-N(2, 5) BP(Y <2)>0.5 -Y-N (-2,5) D Var(X) < Var(Y) 5) Suppose X,Xy..,X, (n>1) is a random sample from N(μ,02) , let-ly, is| then Var(x)- ( Instruction: The followins ass
part iii and iv Exercise 3 (Paired test, known variance of the difference). Let X, Y...
part iii and iv
Exercise 3 (Paired test, known variance of the difference). Let X, Y be RVs. Denote E[X] = Hx and E[Y] = My. Suppose we want to test the null hypothesis Houx = uy against the alternative hy- pothesis H :Mx My. Suppose we have i.i.d. pairs (X1,Y),...,(Xn, Yn) from the joint distribution of (X,Y). Further assume that we know the value of o2 = Var(X-Y). (i) Noting that X1 - Y1,...,xn - Yn are i.i.d. with...
7. (10%) Let f: [0,1] R be defined by _x xe[0,1]n f(x) 0 otherwise Is fe L[0,1]? If yes, find its Lebesgue integral...
7. (10%) Let f: [0,1] R be defined by _x xe[0,1]n f(x) 0 otherwise Is fe L[0,1]? If yes, find its Lebesgue integral. i) Is feR[O,1] ? If yes, find its Riemann integral. ii) ii) What is lim || |, ?
7. (10%) Let f: [0,1] R be defined by _x xe[0,1]n f(x) 0 otherwise Is fe L[0,1]? If yes, find its Lebesgue integral. i) Is feR[O,1] ? If yes, find its Riemann integral. ii) ii) What is lim ||...
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e.,...
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1 with probability 1/2. Show that SZ N(0,1) 5. Let Z N(0, 1) and X = Z2. This distribution is called chi-square with degree of freedom. Calculate P(1 < X < 4) one
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1...
Suppose S= [0,1) with the uniform probability measure and X(s) =s. Calculate MX(z).
Suppose S= [0,1) with the uniform probability measure and X(s) =s. Calculate MX(z).
4. Let Z ~ N(0,1) be a standard normal variable. Calculate the probability (a) P(1 <Z...
4. Let Z ~ N(0,1) be a standard normal variable. Calculate the probability (a) P(1 <Z < 2). (b) P(-0.25 < < < 0.8). (c) P(Z = 0). (d) P(Z > -1).
Q4) Let X and Y be two independent N(0,1) random variable and 10 ei Find the covariance of Z and W.WE3-Y Q4) Let X and Y be two independent N(0,1) random variable and 10 ei Find the covarian...
Q4) Let X and Y be two independent N(0,1) random variable and 10 ei Find the covariance of Z and W.WE3-Y
Q4) Let X and Y be two independent N(0,1) random variable and 10 ei Find the covariance of Z and W.WE3-Y
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i ...
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b) Obtain the marginal pdf of S.
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b)...
Let at XW) =andom variable. Prove III. VARIANCE PROOFS (SINGLE RANDOM VARIABLE) Let S be a...
Let at XW) =andom variable. Prove III. VARIANCE PROOFS (SINGLE RANDOM VARIABLE) Let S be a sample space. Let a, b, c be real numbers. a) Let X(W) = c for all w ES. Prove that Var(X) = 0. b) Let Y be a random variable. Prove that Var(Y + b) = Var(Y). c) Let Z be a random variable. Prove that Var(az) = a-Var(Z). d) Let W be a random variable. Use parts (b)-(c) to prove that Var(aW+b) =...
Exercise 3 Let f be an analytic function on D(0,1). Suppose that f(z) < 1 for...
Exercise 3 Let f be an analytic function on D(0,1). Suppose that f(z) < 1 for all z € C and f() = 0. Show that G) . (Hint: use the function g(z) = f(2).)
CPoisson can not be determined. distribution P(np) ) Suppose X~N(0,1) and YN(24), they are independent, then (is incorrect. DX+Y-N(2, 5) BP(Y <2)>0.5 -Y-N (-2,5) D Var(X) < Var(Y) 5) Suppose X,Xy..,X, (n>1) is a random sample from N(μ,02) , let-ly, is| then Var(x)- ( Instruction: The followins ass
part iii and iv
Exercise 3 (Paired test, known variance of the difference). Let X, Y be RVs. Denote E[X] = Hx and E[Y] = My. Suppose we want to test the null hypothesis Houx = uy against the alternative hy- pothesis H :Mx My. Suppose we have i.i.d. pairs (X1,Y),...,(Xn, Yn) from the joint distribution of (X,Y). Further assume that we know the value of o2 = Var(X-Y). (i) Noting that X1 - Y1,...,xn - Yn are i.i.d. with...
7. (10%) Let f: [0,1] R be defined by _x xe[0,1]n f(x) 0 otherwise Is fe L[0,1]? If yes, find its Lebesgue integral. i) Is feR[O,1] ? If yes, find its Riemann integral. ii) ii) What is lim || |, ?
7. (10%) Let f: [0,1] R be defined by _x xe[0,1]n f(x) 0 otherwise Is fe L[0,1]? If yes, find its Lebesgue integral. i) Is feR[O,1] ? If yes, find its Riemann integral. ii) ii) What is lim ||...
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1 with probability 1/2. Show that SZ N(0,1) 5. Let Z N(0, 1) and X = Z2. This distribution is called chi-square with degree of freedom. Calculate P(1 < X < 4) one
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1...
4. Let Z ~ N(0,1) be a standard normal variable. Calculate the probability (a) P(1 <Z < 2). (b) P(-0.25 < < < 0.8). (c) P(Z = 0). (d) P(Z > -1).
Q4) Let X and Y be two independent N(0,1) random variable and 10 ei Find the covariance of Z and W.WE3-Y
Q4) Let X and Y be two independent N(0,1) random variable and 10 ei Find the covariance of Z and W.WE3-Y
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b) Obtain the marginal pdf of S.
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b)...
Let at XW) =andom variable. Prove III. VARIANCE PROOFS (SINGLE RANDOM VARIABLE) Let S be a sample space. Let a, b, c be real numbers. a) Let X(W) = c for all w ES. Prove that Var(X) = 0. b) Let Y be a random variable. Prove that Var(Y + b) = Var(Y). c) Let Z be a random variable. Prove that Var(az) = a-Var(Z). d) Let W be a random variable. Use parts (b)-(c) to prove that Var(aW+b) =...
Exercise 3 Let f be an analytic function on D(0,1). Suppose that f(z) < 1 for all z € C and f() = 0. Show that G) . (Hint: use the function g(z) = f(2).)
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