Consider a regression setting with input data XER® following a Gaussian distribution X~N(4, E). The target variable y is a scalar, p(y|X) is a Gaussian distribution with mean a"X + b and variance o?. Select one or more true statements from below. a. var,-p, [y] 2 o² b. Ey~p(y), [y] = a"µ + b c. Egxp(x\y)» [X] is a linear function of y d. p(X|y) is a Gaussian distribution e. None of above statements is true
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- 1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.Suppose that Y1,Y2,Y3 denote a random sample from an exponential distribution with density function f(y) = Consider the following four estimators of θ: ?1θe−y/θ, y>0,0, otherwise. θˆ =Y, θˆ =Y1+Y2, θˆ =Y1+2Y2, θˆ =Y1+Y2+Y3 =Y ̄. 11223343 Which estimators are unbiased? Among the unbiased estimators, which has the smallest variance?Let X be a random variable with exponential distribution having lambda=6 and let Y = 3X. a. Find P(X>0.25) b. Find the mgf of Y c. Find the pdf of Y
- A civil engineer is studying the live loads on a highway bridge. Let Y and Z denote the live load (in ten thousands of pounds) for traffic going east and west, respectively, at a randomly selected time of day. Suppose these random variables have a joint distribution with density function f(y,z) = (1/400)yz2e–y/5–z/2, for y > 0, z > 0. If the loads are in opposite directions independent, what should the correlation coefficient and the regression curve of Y on Z be?The moment generating function can be usedto find the mean and variance of the normal distribution.a. Use derivatives of MX(t) to verify that E(X)=meanand V(X) =varianceb. Repeat (a) using RX(t)=ln[MX(t)], and comparewith part (a) in terms of effort.dW is normally distributed, dW has mean zero, dW has variance equal to dt. Parameter other than dw is assumed as constant. We have a representation of the geometric Brownian motion as dS/ S = µ dt + σ dW, prove µ dt + σ dW is normally distributed and find its mean and variance.
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- A snack food manufacturer estimates that the variance of the number of grams of carbohydrates in servings of its tortilla chips is 1.23. A dietician is asked to test this claim and finds that a random sample of 24 servings has a variance of 1.27. At α=0.10, is there enough evidence to reject the manufacturer's claim? Assume the population is normally distributed. (b) Find the critical value(s). I do not know how to calculate the critical value with my calculator. I have a TI83+.Suppose X1, . . . , Xn are i.i.d. from a continuous distribution with p.d.f. fθ(x) = 1/θ if 0 ≤ x ≤ θ, where θ > 0 is an unknown parameter. (a) Find E(X1) (b) Find the MME for θ. (c) Compute the variance of your MME from part (a).A random variable X has probability density function (pdf) fx(x), wherefx(x)= c(1–x4) -1≤x≤1elsewherei)Find the cdf Fx(x)of Xii)Find ciii)Find the variance of X.