Let f(x) (1/4) (1/2)for1,0,1 and f(x) a pmf? If yes, re-express the pmf by a table for other a va...
2. Let X ~ Bin(4, ), i.e., the PMF of X is given by Px(1) = (1) (1) *,* = 0,1,2,3,4. Find Px B(k) where B = {X #0}.
3. Let f(x,y) = xy-1 be the joint pmf/ pdf of two random variables X (discrete) and Y (continuous), for x = 1, 2, 3, 4 and 0 <y < 2. (a) Determine the marginal pmf of X. (b) Determine the marginal pdf of Y. (c) Compute P(X<2 and Y < 1). (d) Explain why X and Y are dependent without computing Cou(X,Y).
2. Let Px(x) = 1, X = 1,2,3, 4, 5, zero elsewhere, be the pmf of X. Find P(X = 1 or 2), P(3 < X < ), and P(1 < X < 2).
ciule jolh! PMF and the marginal PMFs? 6.14 Let X and Y be discrete random variables. Show that the function p: R2 R defined by p(r, y) px(x)pr(y) is a joint PMF by verifying that it satisfies properties (a)-(c) of Proposition 6.1 on page 262. Hint: A subset of a countable set is countable CHAPTER SIX Joindy Discrete Random Variables 6.2 Joint and marginal PMFs of the discrete random variables x numher of bedrooms and momber of bwthrooms of a...
3. Let X1,..Xn be a sample with joint pdf (or pmf) f(x,0), 0 e 0 c R. Suppose that {f(x, 0) 0 e 0} has monotone likelihood ratio (MLR) in T(X,). Consider test function if T(xn)> c if T(xn) c if T(x)<c 0 E [0,1 and c 2 0 are constants. Prove that the power function of ¢(X,) is where non-decreasing in 0 3. Let X1,..Xn be a sample with joint pdf (or pmf) f(x,0), 0 e 0 c R....
that f'(2) is continuous and that F(x) is an antiderivative of f(1). the following table of values: 6 f(x) F() r=0 = 2 r = 4 1 = 6 -2 1 -4 6 2. -3 5 6 -4 2 3 7 (a) Evaluate [u(z) – f(x) – 3)?f'(x)da. b) Evaluate ſz, za f'(x)dx
Problem 4 Let S :R R be such that f (x + y) = f(x) + f(y) for all sy ER Also assume that limf () = LER. 1. Show that f (2x) = 2 (s). 2. Use the result from part 1 to determine the value of L.
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
es F no yes 30 70 no 130 170 Table 1: Relationship between faulty modules (F-yes) and reuse, (R- yes) (cell values are number of modules) i) Produce the joint probability table for P(R, F)
Power function sample with joint pdf (or pmf) f (x |0), 0 e 0 c R. Suppose Let X1,..., X,n be a that {f(xn0) : 0 E 0} has monotone likelihood ratio (MLR) in T(X). Consider test function if T(xn)> c 1 if T(Xn) (Xn) C if T(xn)c 0 where y E [0, 1] and c > 0 are constants. Prove that the power function of ø(X,,) is non-decreasing in 0 sample with joint pdf (or pmf) f (x |0),...