Suppose X~Unif(1,3). Find the p.d.f. of Z = e^X.
Hint: The c.d.f. of Z is G(z) = Pr(Z ≤ z)
= Pr(e^X≤ z)
......
Now you can get the p.d.f.
g(z) = (d/dz)G(z) = ......
Suppose X~Unif(1,3). Find the p.d.f. of Z = e^X. Hint: The c.d.f. of Z is G(z)...
1. Suppose that the p.d.f. of a random variable X is as follows: for 0<x<2, for 0 〈 x 〈 2. r for 0<< f(x) = 0 otherwise. Let Y - X (2 - X). First determine the c.d.f. of Y, then find its p.d.f. (Hint: when computing c.d.f., plotting the function Y- X(2 - X) which may help. )
Suppose the c.d.f. of X is F(t) 3 for 0<t< (a) What is F(5)? (b) What is F(-5)? (c) Compute the p.d.f of X. (d) Compute the mean of X (e) Compute the variance of X. (f) Compute the standard deviation of X (g) Compute the squared coefficient of variation of X.
Pls answer both parts. Thanks! Problem 1A (expectation and p.d.f. of a function of a random variable, Y-g(X)). Consider a random variable X~Uniformle, e2] and define the new random variable Y - In X (a) Compute E(Y), using the theorem that states that E(g(()x() dz. (b) Now we will calculate E(Y) by computing the probability density function of Y first. To do so the initial step is to figure out what the c.d.f. of Y is, by noticing that: Note...
Let, f(x)= 1/15e-x/15, 0≤ x < ∞ be the p.d.f. of X i. find the c.d.f., F(x), for f(x). ii. find the values of µ and ?2. iii. what is the moment generating function? iv. what is the probability that 20<x<40? v. what percentile is µ? vi. what is the value of the 25th percentile?
Find the conditional p.d.f.’s f(y|x) and f(z|x, y). 4. Suppose that random variables (X, Y, Z) have the joint p.d.f. f(x,y,z)-' 0, otherwise . ind the conditional p.d.f.'s f(yx) and f (z x,y
Suppose that U Unif(-2,5) and that Y = g(u) = u? a Find the density of Y, fy(y) b Find E[Y] using this derived density c Find E[U?] using the density of U (should match part b) Hint: draw the problem
(50 points) Suppose that the joint p.d.f. of X and Y is as follows: for x 2 0, y 2 0, and x + y <1 elsewhere 2. 24xy f(x)0 a) Determine the value of P(X < Y). b) Determine the marginal p.d.f.'s for Xand Y c) Find P(X> 0.5) d) Determine the conditional p.d.f. of X|Y = 0.5 e) Find P(X> 0.5|Y 0.5) f) Find P(X> 0.5|Y> 0.5) g) Find Cov (X, Y)
2. Suppose that the p.d.f. of a random variable X is given as in last question. Now let Y 4-X3 Find its p.d.f
Find faults in the following arguments, with brief explanations: (a) First faulty argument: 2(2) (3x) F(x) 3 (3) F(a) (4) F(a)→G(a) 1,3 (5) G(a) 1,3 (6) (Vr) G(x) 1,2 (7) (Vr) G(x) 3,4 MP 2,3,6 3 E (b) Second faulty argument: (1) (yz)(33) H(z, y) 1 (2) y) H(a, y) (3) (4) 5(5) 4.5 (6) 4.5 (7) (3) H(by) H(a, b) H(b, a) H (a, b) ЛНФа) (3y)(H(a,y) ΛΗ(y, a)) 4,5 л! 6ヨ1 Now find models to demonstrate that the...
7. Suppose X,Y "Unif( 0, 0+1). Suppose the scientists wish to test Ho : 0 = 0 versus H: >0. Suppose we have two possible test procedures that we wish to compare: Decision rule for test procedure 1: Reject Ho in favor of H, if Y>0.95. Decision rule for test procedure 2: Reject Ho in favor of H, if X+Y > d. Find the value of d for which the second of these test procedures has the same significance level...