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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Answered: 3. Consider the linear probability…

A First Course in Probability (10th Edition)

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Chapter1: Combinatorial Analysis

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Let f, g be probability densities such supp(f) c supp(g). show yes X1, ... , Xn X1,..., X, iid. L(g) then f(X;) W; = g(X;) is hopeful 1. Argue why the weights need to be re-normalized even when the normalization constants of the densities f and g are known.
9. If X and Y are two random variables and let g(X) be a random variable. Show that (a) E[g(X) X=x] = g(x). (b) E[g(x)Y|X=x] = g(x) E[Y|X=x]. Assume that E[g(x)] and E[Y] exist.
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A random point (X,Y) is distributed uniformly on the square with vertices (1,1), on the square. (1,-1), (-1, 1), and (-1,-1). That is, the joint pdf is f(x,y) Determine the probabilities of the following events. (a) x² + y² 0 (c) |X+Y| < 2
2. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. (a) Find P(X1 = 2 and X2 = 2). (b) Find P(X1 + X2 = 1). (c) Find E(X1 + X2). (d) Find Var(X1 + X2).
If E[X] = 1 and Var(X) = 5, use definition of variance and properties of expectation to find (a) E[(2 + X)^2] (b) V ar(aX) for any constant a. (c) V ar(X + b) for any constant b. (d) V ar(4 + 3X)
Example 18.2 For the probability density of a system of random variables (X, Y): f(x, y) = 0.5 sin (x + y) (0 < x < 7,0 << 7), 2' determine (a) the distribution function of the system, (b) the expectations of X and Y, (c) the covariance matrix.
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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.