**Problem 2.** (SW 11.8(a-c)) Consider the linear probability model $ Y_i = \beta_0 + \beta_1 X_i + u_i $, and assume that $ E(u_i|X_i) = 0 $.(a) Show that $ Pr(Y_i = 1|X_i) = \beta_0 + \beta_1 X_i $.(b) Show that $ \text{var}(u_i|X_i) = (\beta_0 + \beta_1 X_i)[1 - (\beta_0 + \beta_1 X_i)] $. [*Hint:* You will need the formula for the variance of a Bernoulli variable, see, e.g., SW Equation (2.7).](c) Is $ u_i $ heteroskedastic? Explain.

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Answered: Problem 2. (SW 11.8(a-c)) Consider the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Example 3.6. Let X and Y be jointly continuous random variables with joint PDF is given by: 3 fx.x(x, y) =(1+ a²y)!(0,1)(x)/(0.2)() Note that fx(x) = (1+ x²) I0,1) (2) fy(y) : (1+ y) I(0,2)(y) Solve for fx|y(x | y). Remark. It follows immediately from the definition of conditional probability density function that fx,y(r,y) = fxjy(x | y) · fy (y), fx.x (x, y) = fy|x(y | x) · fx(x), -00
0.5 A rental truck costs $45.45 for a day plus 94¢ per mile. (a) If x is the number of miles driven, write an expression for the total cost C, in dollars, of renting the truck for a day.C = (b) Find the total cost of the rental if it was driven 188 miles.C = $
1) Assume a relationship Y = g2(91(X)) between the two random variables X and Y where gi and g2 are two functions. To compute the mean of Y, we can choose any of the following equivalent expressions: E(Y)= | fr (u)ydy = | fn(x)(u)g2(u)du = / fx(#)g2(9n(x)dr (1) -00 where fy (y), f91(x)(u) and fx(x) are the probability density functions (p.d.f.s) of Y, 91(X) and X, respectively. Now consider Y = X? where X is N(0, o2). a) Determine the p.d.f. fy (y) of Y. Hint: for y > 0, fy (y) Prob(Y < y) = Prob(-< X < VI) = % x (x)dx. b) Then evaluate explicitly E(Y)= S fy(y)ydy.
3.2.13 Y is a continuous random variable with { 2(1- y), 0< y < 1; fy(4) = { 0, otherwise. Derive Fy(t).
A coin is tossed twice. Let Z denote the number of heads on the first toss and W the total number of heads on the 2 tosses. If the coin is unbalanced and a head has a 40% chance of occurring, find (a) P( W=2 | Z= 1) (b) P( Z=0 | W = 1) ; (c) are Z and W independent?
A Gaussian random variable has a PDF of the form √ fx(x) = (a) Pr(X>0), (c) Pr(X 3), (g) Pr(X+2] > 1), NT exp(-2(x + 1)²). Write each of the following probabilities in terms of Q-functions (with positive arguments) and also give numerical evaluations: (b) Pr(X>2) (d) Pr(X<-4), (f) Pr(|X+1 <2) (h) Pr(X-1<2)
Do #2
3. The expectation operator E can be applied to a random vector. Specif- ically, if X = [X1 X2 …… Xn]", then EX = [EX1 EX2 .… EXn]". Show that E[(X – EX)(X – EX)"]= [cov(X¡, X;)]"j=1 •
Example 2.1.1 Let S = R, and suppose P is a probability measure on R. Define F(x) by F(x) = P((-∞, x]), XER. (2.3) Then (i) F is right continuous,
If the odds against T occurring are 8:5, then find P(T) and P(T'). P(T) = (Simplify your answer.) P(T') = (Simplify your answer.)
Example 7-36. Let {Xx} be mutually independent, each assuming the values 0.12 . ..., a - 1 with probability . Let S, = X1 + X2 + ... + Xp. Show that the probability generatino (6) = {; P (s) : 1- sa a(1 – s) function of S, is : P (S„ = j) = £ (-1)"*j*av (4)(;=u) and hence %3D - av v=0

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