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Chapter 4 Solutions
Probability and Statistical Inference (9th Edition)
- Suppose X and Y are two independent variables with variance 1. Let Z = X+bY where b > 0. If Cor(Z, Y ) = 1/2, what is the value of b?arrow_forwardLet X is continuous uniform (1,7) and Y is exponential with mean 2. If the correlation coefficient of X and Y is 0.5, find the variance of (X+Y).arrow_forwardLet X and Y be two random variables with Var(X) = 6, Var(Y ) = 3, andthe correlation ρ(X, Y ) = −1/4. Find the value of Var(X − 2Y + 7).arrow_forward
- 9. Which one of the following is the OLS estimator of the variance of slope term for the regression Y = Bo + B1X, + ųę, t = 1,2, ..,T? Here, x = X, – X and y, = Y; – Y. %3D a) of. = (T-2)E-1 x b) o (T-2) E-1 *? c) o (T-2)E-1YẺ d) o Var (X) Var (Y) e) o3, Cov (X,Y) Var (Y)arrow_forward4. Suppose that X has pdf f(x) = 3x² for 0 < x< 1. Find the pdf of the random variable Y = VX.arrow_forwardThe random variables X,Y have variance Var(X)=36 and Var(Y)=1 and their correlation is Cor(X,Y)=−3/4. Calculate Var(X+Y) with a full explanationarrow_forward
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- - (Sec. 6.1) Using a long rod that has length µ (unknown), you are going to lay out a square plot in which the length of each side is µ. Thus the area of the plot will be µ². However, because you do not know the value of µ, you decide to make n independent measurements X1,...,X, of the length. Assume that each X; has mean µ and variance o². (a) Show that X² is not an unbiased estimator for the area of the square plot µ². [Hint: for any rv Y, E[Y²] = V[Y] + E[Y]². Apply this for Y = X.] (b) For what value of k is the estimator X² – kS² unbiased for µ²?arrow_forwardConsider a random sample from NB(r, p) where the parameter r is known to be 3. An experiment is run with n = 10 trials and the sample mean is observed to be x̄ = 0.6. (a) Derive a formula for the MLE p̂ as a function of n, r and X̄ . (b) Find the estimate of p. (c) Find the MLE for the population mean.arrow_forwardThe p.d.f. of a random variable X' is as shown in the figure. The pdf is zero for X 5. Calculate (i) the maximum value of p.d.f. (ii) expectation of X, E(X) (iii) variance of X. fx (x) karrow_forward
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