Suppose the variance of the difference of two random variables X and Y is zero. Then, show that X = Y + c for some constant c almost surely. Also find the value of c.

Select the equation of the least squares line for the data: (51.00, 1.0), (48.75, 2.5), (52.50, .5), (46.50, 5.0), (45.00, 4.5), (41.25, 6.5), (43.50, 5.0). a) ŷ = -28.956 − 0.54067x b) ŷ = 28.956 − 0.59474x c) ŷ = 0.54067x − 28.956 d) ŷ = 31.852 − 0.59474x e) ŷ = 28.956 − 0.54067x f) None of the above

Q3. Which one is not describe the Ordinary Least Squares (OLS) assumptions? A) The regression model should not be non-linear in the parameters B) The regression model is not incorrectly specified C) The number of parameters must be greater than the number of observations. D) Given the value of X, the variance of error is the same for all observations.

A random variable Y has a uniform distribution over the interval (8,, e,). Derive the variance of Y. Find E(Y)? in terms of (e,, e,). E(Y)? = Find E(Y) in terms of (e,, e,). E(Y) = Find V(Y) in terms of (8,, e,). V(Y) =

Suppose X and Y are independent. X has a mean of 1 and variance of 1, Y has a mean of 0, and variance of 2. Let S=X+Y, calculate E(S) and Var(S). Let Z=2Y^2+1/2 X+1 calculate E(Z). Hint: for any random variable X, we have Var(X)=E(X-E(X))^2=E(X^2 )-(E(X))^2, you may want to find EY^2  with this. Calculate cov(S,X). Hint: similarly, we have cov(Z,X)=E(ZX)-E(Z)E(X), Calculate cov(Z,X). Are Z and X independent? Are Z and Y independent? Why? What about mean independence?

The random vector x obtained by adding the vectors x1, x2 and x3 end-to-end has a multivariate Gaussian distribution. How do you find the distribution of x2?

Consider the following simple linear regression. iid Y = Po + B₁X + e, e~ N(0,02), i 1,..., n.

Let X be the Random Variable: x 0 1 2 = x) P(X= 0.5 0.2 0.3 Let X1, X2 be the identically distributed Random Variables so that X1, X2 ~ X. Find the expected value and variance of both X and X1 + X2.

An owner of a home in the Midwest installed solar panels to reduce heating costs. After installing the solar panels, he measured the amount of natural gas used y (in cubic feet) to heat the home and outside temperature x (in degree-days, where a day's degree-days are the number of degrees its average temperature falls below 65° F) over a 23-month period. He then computed the least-squares regression line for predicting y from x and found it to be ŷ = 85 + 16x. The software used to compute the least-squares regression line for the equation above says that  r2 = 0.98.  This suggests which of the following? 1. Gas used increases by square root of 0.98 = 0.99 cubic feet for each additional degree-day?  2. Although degree-days and gas used are correlated, degree-days do not predict gas used very accurately. 3. Prediction of gas used from degree-days will be quite accurate.

ox² = 3, oy² = 5, oxy = 2, Z = 2Y - 4X – 2 a. Determine the variance of Z.