E(X) = 3 λ, E(Y) = 2 λ, and W = k(2X + 3Y), then what is k if W is an unbiased estimator for λ?
x - y + 2z + w = +4 -2x + 3y = -4 x + y + z - 4w = +3 Rewrite linear system as matrix
3Y 2 1. (20 points) Suppose that X and Y independent random variables. Let W 2x (a) Consider the following probability distribution of a discrete random variable X: 12 P(X) 00.7 0.3 X Compute the mean and variance of X (b) Use your answers in part (a). If E(Y)=-3 and V(Y)= 1, what are E(W) and V (W)?
3. Given f(x,y)= sin?(2x+3y?).e***; (a) Find f (x,y). (b) Find f (x,y).
Question 3 1p What is the integrating factor for xy-3y=x^2? x^(-3) -3/x O e^(-3/) O e^(1/2x^2)
x + y + z = 6 2x - y - z=-3 3y - 2z = 0 Question 1 (3 points) 1. X = 3. z = Blank 1: Blank 2: Blank 3: Question 2 (2 points) Picture or screenshot of your answer to #1 (from the matrix calculator). BIU E SÅ S T 2
2. x+4y= 14 2x - y=1 x=2, y=3 3. 5x + 3y = 1 3x + 4y = -6 x=2, y=-3 | 4, 2y- 6x =7 3x - y=9 No solution/Parallel lines
1. xy' = 3y + 3 3. (x + 1)y' - (2x + 3) y = 0
3. Suppose x,y,z satisfy the competing species equations <(6 - 2x – 3y - 2) y(7 - 2x - 3y - 22) z(5 - 2x - y -22) (a) (6 points) Find the critical point (0,Ye, ze) where ye, we >0, and sketch the nullclines and direction arrows in the yz-plane. (b) (6 points) Determine if (0, yc, ze) is stable. (c) (8 points) Determine if the critical point (2,0,0) is stable, where I > 0.
Suppose X~Pois(A) and Y ~Pois(2A) are independent random variables. Consider a linear estimator of λ, that is, λ = aX + bY. (a) Find an expression for the bias of λ, in terms of a and b, and determine a condition on the values of a and b, such that λ is unbiased. (b) Of all the values of a and b that make the estimator unbiased, find the values of oa and b that minimize the variance of the...
4. Co ider dĀ, where R is the parallelogram enclosed by the lines x-3y=0, x-3y=4, 2x-y=2, Å 2x - y and 2x-y=7. Fill in the boxes: Let u=x-3y, and v= 2x - y. Then in terms of u and v, we can set up the PX - 3 ingen i 19 = 3/d2=SHH dvdu. (You do not actually evaluate the integral.) dvdu van de integral as: JJ 2 actually salane te imeni)