the E(X) = a+b/2 when X has a uniform density function, f(x) = 1/b-a, a<x<b. Prove this.
Hence, for uniform distribution E(x) = (a+ b)/2
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the E(X) = a+b/2 when X has a uniform density function, f(x) = 1/b-a, a<x<b. Prove...
Problem 2 Suppose X ~Uniform[0,1 (1) What is the density function? (2) Calculate E(X), E(X2), and Var(X). (3) Calculate F(x)-P(X x) for x E [0, 1]. (4) Let Ylog X. Calculate F(-P(Y 3 y) for y 20. Calculate the density of Y.
Suppose X is a random variable that has density function f(x) = (1/2)e^−|x| for −∞ < x < ∞. Find: (a) (2 pts) P(X < 10). (b) (4 pts) The c.d.f. of X2. (c) (4 pts) V ar(X)
1. The density function of b is given by kx(1 - x) f(x) = { for 0 < x 51, elsewhere. (a) Find k and graph the density function. (b) Find P(1/4 < ſ < 1/2). (c) Find P(-1/2 sã < 1/4). (d) Find the CDF and graph it. (e) Find E( ), E(52), and V(5). 1. The density function of ğ is given by |kx(1 – x) o for 0 < x 51, elsewhere. f(x (a) Find k and...
2. Prove that if X, Y have a joint density, then for any Be B, f(y, x) JB f(x) 2. Prove that if X, Y have a joint density, then for any Be B, f(y, x) JB f(x)
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...
. A random variable X with P(X> 0) 1 has density function f(x) cx299e3. Find: with P(X >0) 1 has density function f (x)cx2s e ) E(X) ) Var(x)
(a) (2 pt) If X is uniform on (0,1), then for what function f is f(x) exponential with parameter 1? (b) (3 pts) If X,Y are independent standard normal random variables N(0,1), what is the density of X -Y?
X is a random variable with density function f(x) = x² /3 for -1 < x < 2,0 else. U is uniform(0,1). Find a function g such that g(U) has the same distribution as X.
A certain random variable X has the probability density function f(x)= e-*+2 for x > 2. Find its variance.
Problem 1-5 1. If X has distribution function F, what is the distribution function of e*? 2. What is the density function of eX in terms of the densitv function of X? 3. For a nonnegative integer-valued random variable X show that 4. A heads or two consecutive tails occur. Find the expected number of flips. coin comes up heads with probability p. It is flipped until two consecutive 5. Suppose that PX- a p, P X b 1-p, a...