Let Y1, Y2, and Y3 be independent, N(0, 1)-distributed random variables, and set X1 = Y1...
Let Y1, Y2, and Y3 be independent, N(0, 1)-distributed random variables, and set X1 = Y1 − Y3, X2 = 2Y1 + Y2 − 2Y3, X3 = −2Y1 + 3Y3.Determine the conditional distribution of X2 given that X1 + X3 = x.
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Let Y1, Y2, and Y3 be independent, N(0, 1)-distributed random
variables, and set X1 = Y1...
Let (X1, Y1) and (X2, Y2) be independent and identically distributed continuous bivariate random variables with joint pr...
Let (X1, Y1) and (X2, Y2) be independent and identically distributed continuous bivariate random variables with joint probability density function: fX,Y (x,y) = e-y, 0 <x<y< ; =0 , elsewhere. Evaluate P( X2>X1, Y2>Y1) + P (X2 <X1, Y2<Y1) .
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 =...
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 = X(1), and Yk = X(k) – X(k-1), 2<k<n. Find the joint pdf of Yı,Y2, ...,Yn. Hint: Note that (Y1,Y2, ...,Yn) = g(X(1), X(2), ..., X(n)), where g is invertible and differentiable. Use the change of variable formula to derive the joint pdf of Y1, Y2, ...,Yn.
Let X1, X2, X3 be independent Binomial(3,p) random variables. Define Y1 = X1 + X3 and Y2 = X2 + X3. Define Z1 = 1 if Y1...
Let X1, X2, X3 be independent Binomial(3,p) random variables. Define Y1 = X1 + X3 and Y2 = X2 + X3. Define Z1 = 1 if Y1 = 0; and 0 otherwise. Define Z2 = 1 if Y2 = 0; and 0 otherwise. As Z1 and Z3 both contain X3, are Z1 and Z3 independent? What is the marginal PMF of Z1 and Z2 and joint PMF of (Z1, Z2) and what is the correlation coefficient between Z1 and Z2?
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y;...
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3. 1. What is the distubution for each Yį, i = 1, 2, 3? 2. What is the expected value for Y1 + Y2 +Yz? 3. Are Yį and Y2 independent? Explain it by computing their joint P.M.F. 4. What is the variance of Y1...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a)...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
Set A contains three numbers x1, x2 and x3. Set B contains four numbers y1, y2,...
Set A contains three numbers x1, x2 and x3. Set B contains four numbers y1, y2, y3 and y4. These two sets have the following characteristics: Set Mean Standard deviation A 10 2 B 45 5 Set X consists of the following eight numbers: u1 =70, u2 =7x1, u3 =7x2, u4 =7x3, u5 =2y1, u6 =2y2, u7 =2y3, u8 =2y4. Find the mean and standard deviation of the numbers in set X.
Let Y1, Y2, . .. , Yn be independent and identically distributed random variables such that...
Let Y1, Y2, . .. , Yn be independent and identically distributed random variables such that for 0 < p < 1, P(Yi = 1) = p and P(H = 0) = q = 1-p. (Such random variables are called Bernoulli random variables.) a Find the moment-generating function for the Bernoulli random variable Y b Find the moment-generating function for W = Yit Ye+ … + . c What is the distribution of W? 1.
15. (30 points) Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size n = 4 fr...
15. (30 points) Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size n = 4 from a distribution with p.d.f.f(x) 2x, 0 < x < 1, zero elsewhere. Evaluate E[Yalyj]. [Hint: First find the joint p.d.f. of Y3 and Y4, and then find the conditional p.d.f. of Y4 given Y3 y3]
15. (30 points) Let Y1
3. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) =...
3. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) = 2. o if 0<x< 1 else Find EY] where Y = min{X1, X2, X3, X4}.
Consider two independent random samples, X1, X2, X3, X1 and Y1, Y , Y3, 74, Y5,...
Consider two independent random samples, X1, X2, X3, X1 and Y1, Y , Y3, 74, Y5, Y, each from the same population having unknown mean and unknown variance ,2. Consider the set of estimators for p given by S{A} = {ña :fla = (1 - a) X+ay, for 0 <a<1}. What is the value of a, denoted by a*, such that file has the lowest mean square error of all available estimators in S{n} ? Answer:
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 = X(1), and Yk = X(k) – X(k-1), 2<k<n. Find the joint pdf of Yı,Y2, ...,Yn. Hint: Note that (Y1,Y2, ...,Yn) = g(X(1), X(2), ..., X(n)), where g is invertible and differentiable. Use the change of variable formula to derive the joint pdf of Y1, Y2, ...,Yn.
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3. 1. What is the distubution for each Yį, i = 1, 2, 3? 2. What is the expected value for Y1 + Y2 +Yz? 3. Are Yį and Y2 independent? Explain it by computing their joint P.M.F. 4. What is the variance of Y1...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
Let Y1, Y2, . .. , Yn be independent and identically distributed random variables such that for 0 < p < 1, P(Yi = 1) = p and P(H = 0) = q = 1-p. (Such random variables are called Bernoulli random variables.) a Find the moment-generating function for the Bernoulli random variable Y b Find the moment-generating function for W = Yit Ye+ … + . c What is the distribution of W? 1.
15. (30 points) Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size n = 4 from a distribution with p.d.f.f(x) 2x, 0 < x < 1, zero elsewhere. Evaluate E[Yalyj]. [Hint: First find the joint p.d.f. of Y3 and Y4, and then find the conditional p.d.f. of Y4 given Y3 y3]
15. (30 points) Let Y1
3. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) = 2. o if 0<x< 1 else Find EY] where Y = min{X1, X2, X3, X4}.
Consider two independent random samples, X1, X2, X3, X1 and Y1, Y , Y3, 74, Y5, Y, each from the same population having unknown mean and unknown variance ,2. Consider the set of estimators for p given by S{A} = {ña :fla = (1 - a) X+ay, for 0 <a<1}. What is the value of a, denoted by a*, such that file has the lowest mean square error of all available estimators in S{n} ? Answer:
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