Let X be a random variable taking three values: P(X = a₁) = P₁, P(X=a₂) = P2₁ P(X=03) = P3, where p₁ + P2 + P3 = 1 and P₁, P2, P3 € (0, 1). Let A = {X = a₁} and G = {2, 0, A, Ac}. Prove that E (X³|G) = a1₁ + 0²0² +0²P³ 14². P2 + P3

Q: Let X, Y be independent Uniform (0, 2) random variables. Define Y, if X }. Z = Compute E[Z] and…

A: X,Y ~ Uniform(0,2) E(X) = E(Y) = 0+22=1 Var(X) = Var(Y) = (2-0)212=412=13 Z = Yif X <122Yif X≥12…

Q: Suppose that X and Y are random variables with E(X) = 2, E(Y) = 5 and E(X?) = 8, E(Y?) = 30 and…

A: Solution: From the given information,

Q: 5.87 Suppose that Yı and Y2 are independent x² random variables with vị and v2 degrees of freedom,…

A: Given that; Y1  and Y2  are independent random variables and follow chi - square distribution with…

Q: Let X and Y be two independent random variables satisfying first success d bution FS(p). (a) (b)…

A: From the given information, X and Y are two independent random variables satisfying first success…

Q: Let X and Y be two random variables and let r, s, t, and u be real numbers. a. Show that Cov(X+s,…

A: Given: X and Y be two random variables and let r, s, t, and u be real numbers.  We have to show that…

Q: Let x and y be two independent N(0, 1) random variables and Z = 1+ X + XY², W = 1+ X. Find Coy(7,W).

A:

Q: Let X and Y be random variables with finite variance. Prove that |ρ(X, Y)| = 1 implies that there…

A: Let's assume that |ρ(X, Y)| = 1, which means that the correlation coefficient between X and Y is…

Q: If X and Y are two random variables, then the covariance of x+a,Y +b,where a and b are constant is…

A:

Q: (A) ơy = 0xı + Ox2 Vox1 + Ox2 +o? = Ox1 +0x2 (B ) συ .2 (C) oy = -2 x2 X1 .2 -2 (D) Oy/y = X2

A: The standard deviation is not directly added together but the variance is added for variables. So…

Q: If X is a random variable, prove that Cov(X,X) = σX².

A: Let X is a random variable.

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A: From the given information, A person is well with probability 1-p A person is ill with probability…

Q: 3. If X and Y are independent uniform (0, 1) random variables, show that 2 for a > 0 (a + 1)(a + 2)…

A: X and Y are independent U(0,1) random variables:Objective: Show that:

Q: c) Suppose that X and Y are jointly continuous random variable - x f(x,y) = {o* for 0< x <1 and 1< y…

A:

Q: Suppose that the joint CDF of the random variables X and Y is given by 0, x < -2 or y < –5

A: Given that Fx,y=0, x&lt;-2 or y&lt;-538, -2≥x&lt;2, -5≤y&lt;312, x≥2 and -5≤y&lt;312, -2≤x&lt;2 and…

Q: Let U and V be uncorrelated random variables. Consider U = X + Y and V = X. a. Find Cov(U,V). b. Is…

A: Given that  Let U and V are uncorrected  U=X+Y and  V=X  We have to find  a..Cov(U,V)  b..is there…

Q: Find the PMF of X. Find the expectations E[X] and E[XY] Find the variance of X.

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer three sub-parts at a…

Q: 2. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. d.…

A:

Q: If X and Y are two independent random variables, then E(XY) = E(X) E(Y)

A: We need to prove, E(XY)=E(X).E(Y) It is given that X and Y are independent random variables,

Q: 17. Let X1, X2, and X3 be independent, U (0, 1)-distributed random variables. Compute P(X(3) > 1/2|…

A:

Q: If X and Y have the joint probability distributionf(−1, 0) = 0, f(−1, 1) = 1 4 , f(0, 0) = 16 , f(0,…

A: The joint probability distribution of the random variables is given as follows:

Q: Let X and Y be random variables with Var (X₂) o and Var(Y₂) o for all ie {1,...,n}. Assume that each…

A: (a) The covariance between Xi and Yi is given by: Cov(Xi, Yi) = E[(Xi - E[Xi])(Yi - E[Yi])] Since…

Q: 3. Let X and Y be two independent Bernoulli (0.5) random variables. Let Z = X + Y. Show that Var (Z)…

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Q: Let X1, X2, ... Xn random variables be independent random variables with a Poisson distribution…

A: Let x1 , x2 ......xn random variables be independent random variables with a Poisson distribution…

Q: Let X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) +…

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Q: what is E[XY]?

A:

Q: Let X1, X2, X3 form a random sample from U(0,1). Find a- The distribution of the second order…

A:

Q: 4. Let X₁, i = 1, 2, ...., be independent random variables with P(X₁ = 1) = ½, P(X₁ = − 1) = 1/2.…

A: From the given information, Xi, i=1,2,.... be independent random variables with PXi=1=12, PXi=-1=12…

Q: If X1 and X2 are independent random variables with distribution given by P[Xi] = −1 = P[Xi = 1] =…

A: Construct the PMF of X1X2, So, P(X1X2 = x) = P(X1=1, X2 = 1) +P(X1=-1, X2 = -1)    if x =1P(X1=-1,…

Q: Let V=X+Y and U=X-Y are random variables, then the condition which make the covariance

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Q: Show that (a) Cov(aX, bY)= ab Cov(X,Y). (b) Cov(X+a, Y + b) = Cov(X,Y). (c) Cov(X, aX+b) = a Var(X).

A: Let (X ,Y) be a bivariate random variable and a, b be real constants. Covariance measures joint…

Q: For the random variable X with the pmf f (x) = (x² + 4) , P(X = -1| a < 0) x = -2, –1,0, 1,2. Find…

A: Given Information: f(x) = 130x2+4 x = -2, -1, 0 , 1 , 2

Q: Let X1,X2,· · ,Xn be i.i.d. random variables, where X; ~ Bernoull Y1 = X1X Y, = X2X %3D Yn-1 = X_ Yn…

A: A random variable X is called to follow the Bernoulli distribution if it takes only non-negative…

Q: Assume that the random variables X1 and X2 are bivariate normally distributed with mean µ and…

A:

Q: Let X1, …, Xn be independent lognormal random variables and let a1, …, an be constants. Show that…

A:

Q: Let Z be a discrete random variable withE(Z)=0. Does it necessarily follow that E(Z3)=0? If yes,…

A: Solution

Q: Let X,, X2, .. .be independent Cauchy random variables, each with PDF d f(x)= T(d² +x²)' Show that…

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Q: Let X = [X1 X2 X3 X4], where the X; are all equal-mean unit-variance independent random variables.…

A: The complete solution is in given below

Q: Suppose we have three independent random variables X, Y and Z where... Var(X + 2Y) = 13, Var(2Y +…

A:

Q: 17. Let I(W₁₁ W₂) =W₁-W₂. Compute Cov(X, Y). Are they independent? die die = {(W₁, W₂) : w₁ = 1, 2,…

A: Given Ω=Ωdie×Ωdie=ω1,ω2 : ω1=1,2,3,4,5,6 ; ω2=1,2,3,4,5,6

Q: Let X1 and X2 be independent random variables for which P(Xi = 1) = 2/5 and P(Xi = 2) = 3/5 . Define…

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Q: Suppose that W is a random variable with E(W4)&lt; ∞. Show thatE(W2) &lt; ∞.

A: Given: Jensen’s inequality says that if g(.) is a convex function: