A random variable X has a Par(3) distribution, so with distribution function F with F(x) = 0 for x < 1 and F(x) = 1 - x^3 for x >-. For details on the Pareto distribution. Describe how the constant X from a U(0,1) random variable.
A random variable X has a Par(3) distribution, so with distribution function F with F(x) =...
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.
2. Determine whether the function f(x) is a valid probability distribution (PMF) for a random variable with the range 0,1,2,3,4 12 f(x) = 30 3. Suppose X is a random variable with probability distribution (PMF) given by f( and a range of 0,1, 2. Find the distribution function (CDF) for X 6
Q2. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function F and density f. Let b>0. (a) Write the formula for EXI(X < . (b) Apply the general formula from (a) to Pareto distribution with parameter α > 0.
C2.1 (Probability integral transform.) Let X be a random variable with cu mulative distribution function F, and suppose that F is continuous and strictly increasing on R. (i) Show that F has a well-defined inverse function G : (0,1) → R, which is (ii) Using G, or otherwise, show that the random variable F(X) is uniformly 시 strictly increasing distributed on [0,1
A random variable X has a distribution with probability function f(x) = K(nx)2x for x = 0,1,2,...,n where n is a positive integer. a. Find the constant k. b. Find the expected value M(S) = E(esX) as a function of the real numbers s. Compare the values of the derivative of this function M'(0) at 0 and the expected value of a random variable having the probability function above. c. What distribution has probability function f(x)? Let X1, X2 be independent random variables both...
12. (15 points) Let X be a continuous random variable with cumulative distribution function **- F() = 0, <a Inx, a < x <b 1, b<a (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)
help
a random variable X has density function f(x) = cx2 for 0<x<3 and f(x)= 0 others. a. Find constant value o b. Find probability P(1 < X < 2)
Exercise 3.37. Suppose random variable X has a cumulative distribution function F(x) = 1+r) 720 x < 0. (a) Find the probability density function of X. (b) Calculate P{2 < X <3}. (c) Calculate E[(1 + x){e-2X].
(15 points) Let X be a continuous random variable with cumulative distribution function F(x) = 0, r <α Inr, a< x <b 1, b< (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)
Q3. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx Let b> o. (a) Find the cumulative distribution function of Y = (X-b)+ b) Apply the general formula from (a) to Pareto distribution with parameter a > 0. Hint: Consider separately cases b e (0, 1 and b> 1