Question

In: Statistics and Probability

. Let X1,X2,and X3 be three random variables with means, variances, and correlation coefficients, denoted by...

. Let X1,X2,and X3 be three random variables with means, variances, and correlation coefficients, denoted by μ1, μ2, μ3; σ² 1,σ² 2,σ² 3; and ρ12, ρ13, ρ23, respectively.

For constants b2 and b3, suppose

E(X1−μ1|x2, x3) = b2(x2−μ2)+b3

Determine b2 and b3 in terms of the variances and the correlation coefficients

Solutions

Expert Solution


Related Solutions

Let X1, X2, X3 be continuous random variables with joint pdf f(X1, X2, X3)= 2 if...
Let X1, X2, X3 be continuous random variables with joint pdf f(X1, X2, X3)= 2 if 1<X1<2 -1<X2<0 -X2-1<X3<0                         0 otherwise Find Cov(X2, X3)
let X1, X2, X3 be random variables that are defined as X1 = θ + ε1...
let X1, X2, X3 be random variables that are defined as X1 = θ + ε1 X2 = 2θ + ε2 X3 = 3θ + ε3 ε1, ε2, ε3 are independent and the mean and variance are the following random variable E(ε1) = E(ε2) = E(ε3) = 0 Var(ε1) = 4 Var(ε2) = 6 Var(ε3) = 8 What is the Best Linear Unbiased Estimator(BLUE) when estimating parameter θ from the three samples X1, X2, X3
Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2...
Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y = (Y1,Y2,Y3)′ using : Multivariate normal distribution properties.
Let X1, X2, X3, . . . be independently random variables such that Xn ∼ Bin(n,...
Let X1, X2, X3, . . . be independently random variables such that Xn ∼ Bin(n, 0.5) for n ≥ 1. Let N ∼ Geo(0.5) and assume it is independent of X1, X2, . . .. Further define T = XN . (a) Find E(T) and argue that T is short proper. (b) Find the pgf of T. (c) Use the pgf of T in (b) to find P(T = n) for n ≥ 0. (d) Use the pgf of...
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.
Consider a sequence of random variables X0, X1, X2, X3, . . . which form a...
Consider a sequence of random variables X0, X1, X2, X3, . . . which form a Markov chain. (a) Define the Markov property for this Markov chain both in words and using a mathematical formula. (b) When is a Markov chain irreducible? (c) Give the definition for an ergodic state.
Let X1, X2, X3, and X4 be a random sample of observations from a population with...
Let X1, X2, X3, and X4 be a random sample of observations from a population with mean ? and variance ?2. Consider the following two point estimators of ?: b1= 0.30 X1 + 0.30 X2 + 0.30 X3 + 0.30 X4 and b2= 0.20 X1 + 0.40 X2 + 0.40 X3 + 0.20 X4 . Which of the following constraints is true? A. Var(b1)/Var(b2)=0.76 B. Var(b1)Var(b2) C. Var(b1)=Var(b2) D. Var(b1)>Var(b2)
Let the independent random variables X1, X2, and X3 have binomial distributions with parameters n1=3, n2=5,...
Let the independent random variables X1, X2, and X3 have binomial distributions with parameters n1=3, n2=5, n3=2 and the same probabilitiy of success p = 2/5. Find P(X1=1-X3). Find P(X1=X3). Find P(X1+X2+X3>=1). Find the expected value and variance for X1+X2+X3.
Let X1, X2, . . . be iid random variables following a uniform distribution on the...
Let X1, X2, . . . be iid random variables following a uniform distribution on the interval [0, θ]. Show that max(X1, . . . , Xn) → θ in probability as n → ∞
Let X1, X2, . . . be a sequence of independent and identically distributed random variables...
Let X1, X2, . . . be a sequence of independent and identically distributed random variables where the distribution is given by the so-called zero-truncated Poisson distribution with probability mass function; P(X = x) = λx/ (x!(eλ − 1)), x = 1, 2, 3... Let N ∼ Binomial(n, 1−e^−λ ) be another random variable that is independent of the Xi ’s. 1) Show that Y = X1 +X2 + ... + XN has a Poisson distribution with mean nλ.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT