Suppose that random variable X 0 = (X1, X2) is such that E[X 0 ] =...

Suppose that random variable X 0 = (X1, X2) is such that E[X 0 ] = (µ1, µ2) and var[X] = σ11 σ12 σ12 σ22 . (a matrix)

(i) Let Y = a + bX1 + cX2. Obtain an expression for the mean and variance of Y .

(ii) Let Y = a + BX where

a' = (a1, a2) B = b11 b12 0 b22 (a matrix).

Obtain an expression for the mean and variance of Y .

(ii) Suppose Y = a + bX1 + cX2 and Z = d + eX2. Calculate cov(Y, Z).

Expert Solution

orchestra answered 2 years ago

2.2.8. Suppose X1 and X2 have the joint pdf f(x1, x2) = " e−x1 e−x2 x1...

2.2.8. Suppose X1 and X2 have the joint pdf f(x1, x2) = " e−x1 e−x2 x1 > 0, x2 > 0 0 elsewhere . For constants w1 > 0 and w2 > 0, let W = w1X1 + w2X2. (a) Show that the pdf of W is fW (w) = " 1 w1− w2 (e−w/w1 − e−w/w2) w > 0 0 elsewhere . (b) Verify that fW (w) > 0 for w > 0. (c) Note that the pdf fW...

Consider the random sample X1, X2, ..., Xn coming from f(x) = e-(x-a) , x >...

Consider the random sample X1, X2, ..., Xn coming from f(x) = e-(x-a) , x > a > 0. Compare the MSEs of the estimators X(1) (minimum order statistic) and (Xbar - 1) Will rate positively

Consider independent random variables X1, X2, and X3 such that X1 is a random variable having...

Consider independent random variables X1, X2, and X3 such that X1 is a random variable having mean 1 and variance 1, X2 is a random variable having mean 2 and variance 4, and X3 is a random variable having mean 3 and variance 9. (a) Give the value of the variance of X1 + (1/2)X2 + (1/3)X3 (b) Give the value of the correlation of Y = X1- X2 and Z = X2 + X3.

Does a continuous random variable X exist with E(X − a) = 0, where a is...

Does a continuous random variable X exist with E(X − a) = 0, where a is the day of your birthdate? If yes, give an example for its probability density function. If no, give an explanation. (b) Does a continuous random variable Y exist with E (Y − b) 2 = 0, where b is the month of your birthdate? If yes, give an example for its probability density function. If no, give an explanation.

If X is a random variable with CDF F(x) = e-e(µ-x)/β, where β > 0 and...

If X is a random variable with CDF F(x) = e-e(µ-x)/β, where β > 0 and -∞ < µ, x < ∞; calculate the median of X. Also, obtain the PDF of X.

Suppose that X1 and X2 are two random variables. Suppose that X1 has mean 1 and...

Suppose that X1 and X2 are two random variables. Suppose that X1 has mean 1 and variance 4 while X2 has mean 3 and variance 9. Finally, suppose that the correlation between X1 and X2 is 3/8. Denote Y = 2X1 − X2. (67) The mean of Y is (a) 1 (b) 4 (c) -2 (d) -1 (68) The variance of Y is (a) 25 (b) 4 (c) 9 (d) 16 (69) The standard deviation of Y is (a) 5...

Let X be a random variable with CDF F(x) = e-e(µ-x)/β, where β > 0 and...

Let X be a random variable with CDF F(x) = e-e(µ-x)/β, where β > 0 and -∞ < µ, x < ∞. 1. What is the median of X? 2. Obtain the PDF of X. Use R to plot, in the range -10<x<30, the pdf for µ = 2, β = 5. 3. Draw a random sample of size 1000 from f(x) for µ = 2, β = 5 and draw a histogram of the values in the random sample...

Suppose that y = x2, where x is a normally distributed random variable with a mean

Suppose that y = x2, where x is a normally distributed random variable with a mean and variance of µx = 0 and σ2x = 4. Find the mean and variance of y by simulation. Does µy = µ2x? Does σy = σ2x? Do this for 100, 1000, and 5000 trials.

Let X1 and X2 have the joint pdf f(x1,x2) = 2 0<x1<x2<1; 0. elsewhere (a) Find the...

Let X1 and X2 have the joint pdf f(x1,x2) = 2 0<x1<x2<1; 0. elsewhere (a) Find the conditional densities (pdf) of X1|X2 = x2 and X2|X1 = x1. (b) Find the conditional expectation and variance of X1|X2 = x2 and X2|X1 = x1. (c) Compare the probabilities P(0 < X1 < 1/2|X2 = 3/4) and P(0 < X1 < 1/2). (d) Suppose that Y = E(X2|X1). Verify that E(Y ) = E(X2), and that var(Y ) ≤ var(X2).

Let X1 and X2 be a random sample from a population having probability mass function f(x=0)...

Let X1 and X2 be a random sample from a population having probability mass function f(x=0) = 1/3 and f(x=1) = 2/3; the support is x=0,1. a) Find the probability mass function of the sample mean. Note that this is also called the sampling distribution of the mean. b) Find the probability mass function of the sample median. Note that this is also called the sampling distribution of the median. c) Find the probability mass function of the sample geometric...

Question