3. Consider the linear probability model Y₁ = Bo+B₁Xi+ui, where Pr(Y₁ = 1|Xi) = Bo +3₁ Xi. (a) Show that E(u₂|X₂) = 0. (b) Show that Var(u₁|X;) = (Bo + B₁X;)[1 − (Bo + B₁X;)]. (c) Is ui conditionally heteroskedastic? Is u heteroskedastic? (d) Derive the likelihood function.
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![3. Consider the linear probability model Y₁ = Bo+B₁Xi+ui, where Pr(Y₁ =
1|Xi) = Bo +3₁ Xi.
(a) Show that E(u₂|X₂) = 0.
(b) Show that Var(u₁|X;) = (Bo + B₁X;)[1 − (Bo + B₁X;)].
(c) Is ui conditionally heteroskedastic? Is u heteroskedastic?
(d) Derive the likelihood function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2e155de-a132-4806-a3d9-1f08bdb62da3%2Fec27d685-cbf4-4e23-9d29-74375fa3f642%2Flnsn73a_processed.png&w=3840&q=75)
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.The index model has been estimated for stocks A and B with the following results: RA = 0.03 + 0.8RM + eA. RB = 0.01 + 0.9RM + eB. σM = 0.35; σ(eA) = 0.20; σ(eB) = 0.10. The covariance between the returns on stocks A and B is A) 0384. B) 0.0406. C) 0.0882. D) 0.0772. E) 0.4000. 2) Analysts may use regression analysis to estimate the index model for a stock. When doing so, the slope of the regression line is an estimate of A) the α of the asset. B) the β of the asset. C) the σ of the asset. D) the δ of the asset. Choose correct answer with justification.Suppose X is a random variable, whose pdf is defined as follows: 2x = (²x) (u(x) - u(x − 3)) where u(x) is the unit step function. Determine the conditional pdf fx(x 1Let X1, X2, ..., X, be independent random variables and Y = min{X1, X2, ..., Xm}. Fy (y) = 1 – || (1 – Fx,(y)) i=1 (a) A certain electronic device uses 5 batteries, with each battery to have a life that is exponentially distributed with mean of 48 hours and is independent of the life of other batteries. If the device fails as soon as at least one of its batteries fail, what is the expected life of the device?(a) Use the rules of expected value to show that Cov(ax + b, cY + d) = ac Cov(X, Y). Cov(ax + b, CY + d) = E - E(ax + b)E ])-) + adx + bcY + bd - (aE(X) + b) CE = = acE + adE(X) + bcE(Y) + bd + adE(X) + bcE = acE acE(X)E 1)) = — асCov(X, Y) (b) Use part (a) along with the rules of variance and standard deviation to show that Corr(ax + b, cY + d) = Corr(X, Y) when a and c have the same sign. co( b, cY + Corr(ax + b, cY + d) = Oax + bº cY + d Cov(X, Y) la||clox0y Corr(X, Y) lallc| = Corr(X, Y) (c) What happens if a and c have opposite signs? ac O When a and c differ in sign, ac is negative and = -1. Therefore, Corr(ax + b, CY + d) = -Corr(X, Y). la||c| ac O When a and c differ in sign, ac is positive and = 1. Therefore, Corr(ax + b, cY + d) = -Corr(X, Y). Jal|c ас O when a and c differ in sign, ac is positive and = 1. Therefore, Corr(ax + b, cY + d) = Corr(X, Y). lal|c ac O when a and c differ in sign, ac is positive and = -1. Therefore, Corr(ax + b, cY + d) = -Corr(X, Y). la||c| ас O…There are only two states of the world, when a person is well with probability (1-p) and ill with probability p, where (1-p)= 1/3 and p = 2/3. Consider Adam who has utility function U = (Y1, Y2,1 – p,p) = Y,"-P)Y?, where Y; is the income and i =1 is well and i = 2 is ill. When Adam is well, he earns $1000, but when he is ill, he losses $300 in health expenditures and earnings such that Y2 = $700. What is the maximum total premium that Adam is still willing-to-pay? (1-р).X is a continuous random variable having p.d.f. f(x) such that f(x) = kr Osx55 otherwise Find : (a) the constant k; (b) P(15XS 2); (c) (d (e) (f) E(X); E(3X + 1); E(X?); Var (X); (g) Var (4X - 5);Let X and Y be independent, taking on {1,2,3,4,5} with equal probabilities. E((X+Y)2)=?The diameter of a round rock in a bucket, can be messured in mm and considered a random variable X in f(x) f(x) = k(x-x4) if 0 ≤ x ≤ 1f(x) = 0 otherwise. Find the expectation E(X) and variance V(X)Let X1, X2, . . . are independent indicator variables with different probabilities of success. That is, P(Xi = 1) = pi. Define Yn = X1 + X2 + . . . + Xn. Find E(Yn), V ar(Yn) and coefficient of variation of YnCalculate the conditional expectation of Y when X = 2. E(YX2)II. Let X be a discrete random variable with possible values xi, i=1,2,.., and p(xi)=P(X=xi) is called the probability distribution function of X. We have > p(x,)=. The expected value, also called expectation, average, or mean, of X is defined as u= E(X)=, For any function g(x), x=1,2,3,.., E(g(x)) =. . The variance of X: Var(X)= Since we have E(X – µ)² = E(X² – 2µX + µ) = EX² – 2µEX + µ² = EX² – (EX)² the variance of X: Var(X)=.SEE MORE QUESTIONS
