from above E(X22*(2X1-X3)2)=E(X22*(4X12+X32-4X1X3))
=4*E(X22)*E(X1)2+E(X22)*E(X3)2-4E(X22)*E(X1)*E(X3)
=4*1*1+1*1-4*1*0*0=5
5. Suppose that X X', and X are independent random variables such that ELX.]-O and BLX?)...
5. Suppose that X1, X2, and X3 are independent random variables such that E[Xl i = 12,3. Find the value of E(X 2x1-X3)2]. mt random variables such that EX-0 and ER-i for 0 and ELX 1 for
5. Suppose that Xi, X2, and X3 are independent random variables such that EX i12,3. Find the value of ElX(2X1- X3)21. dEX? 1 for
Suppose that the standard normal random variables X and Y are independent. Find P(0 < X<Y). 8 O 1 4T 0 1 8л Ala
Suppose that random variables X and Y are independent. Further, X is an exponential random variable with parameter 1 = 3, and Y is an uniformly distributed random variable on the interval (0,4). Find the correlation between X and Y, rounded to nearest .xx
Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is Y 1Xi Find/prove the mgf of Y and find E(Y), Var(Y), and P (8 Y 9) if a) X1,.,X4 are Poisson random variables with means 5, 1,4, and 2, respectively. b) [separately from part a)] X,., X4 are Geometric random variables with p 3/4. i=1
3. Suppose that X and Y are independent exponentially distributed random variables with parameter λ, and further suppose that U is a uniformly distributed random variable between 0 and 1 that is independent from X and Y. Calculate Pr(X<U< Y) and estimate numerically (based on a visual plot, for example) the value of λ that maximizes this probability.
Suppose that X and Y are independent standard normal random variables. Show that U = }(X+Y) and V = 5(X-Y) are also independent standard normal random variables.
5. Suppose that the joint pdf of the random variables X and Y is given by - { ° 0 1, 0< y < 1 f (x, y) 0 elsewhere a) Find the marginal pdf of X Include the support b) Are X and Y independent? Explain c) Find P(XY < 1)
2. Let X and Y be jointly Gaussian random variables. Let ElX] = 0, E[Y] = 0, ElX2-4. Ey2- 4, and PXY = [5] (a) Define W2x +3. Find the probability density function fw ( of W. [101 (b) Define Z 2X - 3Y. Find P(Z > 3) 5] (c) Find E[WZ], where W and Z are defined in parts (a) and (b), respectively.
Question 7 (1 point) Suppose we have the following values for variables X (independent) and Y (dependent) IX-X)(y-7) --630 Z(X - x)2 = 168 SSE = 1225 SSR = 2362.5 What is the value of R-Square (round to four decimal places)? Question 8 (1 point) Suppose we have the following values for variables X (independent) and Y (dependent) EIX-XIY-7)=-630 IIX Xj2 = 168 SSE = 1225 SSR = 2362.5 What is the value of the slope coefficient for the simple...