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1. Consider the following two probability density functions: f(3) = 2053 } for a <I<02 and...
1. Consider two probability density functions on [0,1]: f0(x) = 3x2 and f1(x) = 4x3. a.) Construct the most powerful test for H0 : X ~ f0 against HA : X ~ f1 with the significance level alpha = 0.1 b.) Find its power.
3.5. Suppose that X and Tare independent, continuous random variables and that U-X+1. Denote their probability density functions by f(x), g(y) and h(u) and the corresponding cumulative probability functions by F(x), G(2) and H(u) respectively. Then For a fixed value of I, say T-y,this probability is F(u-), and the probability that I will lie in the range y to y+dy is g()dy. Hence the probability that Usu and that simultaneously Y lies between y and y+dy is F(u-)go)dy and so...
1. Consider the following distribution of (X Y) where X and Y ae both binary random variables: 1/4 i (a)-(0.0 1/4 if (x, y) (0,0) 1/8 if (r,y) (1,0) Jx3/8 if (r,)- (0,1) ,Y (z, y) = 1/4 if (, ) (11 (a) What is the probability density function of Y? (b) What is the expectation of Y1 (c) What is the variance of Y? (d) What is the standard deviation of Y? (e) Do the same to X. (f)...
stats (6) Consider the following joint probability density function of the random variables X and f(x,y) = 9, 1<x<3, 1<y< 2, elsewhere. (a) Find the marginal density functions of X and Y. (b) Are X and Y independent? (c) Find P(X > 2).
QUESTION 4 Suppose Xis a random variable with probability density function f(x) and Y is a random variable with density function f,(x). Then X and Y are called independent random variables if their joint density function is the product of their individual density functions: x, y We modelled waiting times by using exponential density functions if t <0 where μ is the average waiting time. In the next example we consider a situation with two independent waiting times. The joint...
7.2. Which of the following functions represent a probability density function for a continuous random variable? Hint: Check if both rules of a proper probability density function hold. (a) f(z) = 0.25 where 0-1-8. b) f(r) =1/2 where 0 <1<2
As described in class, the Poisson Bracket [F, G] between two functions Fand G of the generalized positions q, and momenta pi is defined as: Consider a system with Hamiltonian H-P2/2m-Vr = (P, 2+py 2+pz2y2m)-y(x"2 + y"2 + z ^2)-U2 where yis a constant. a) Evaluate [Lz, H] and interpret the result in two ways i.e. what it says about L, and what it says about H b) Using the Poisson Bracket and the given Hamiltonian, find the value of...
2. Suppose that (X,Y) has the following joint probability density function: f(x,y) = C if -1 <r< 1 and -1 <y<1, and 0 otherwise. Here is a constant. (a) Determine the value of C. (b) Are X and Y independent? (Explain why or why not.) (c) Calculate the probability that 2X - Y > 0 (d) Calculate the probability that |X+Y| < 2 3. Suppose that X1 and X2 are independent and each is standard uniform on (0,1]. Let Y...
8. Let 02 (1)e, 00; x > 0 -0x f(x) = 1 +0 (a) Show that f(x) is a probability density function (b) Find P(X > x) (c) Find the failure rate function of X
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...