Let X be some continuous random number whose density function is continuous and everywhere positive: fx(x) > 0 for all x ∈ R. Let FX be the accumulation function of this distribution. Show that the distribution of the transformation FX(X) is a continuous uniform distribution on the interval (0,1). Hint: You might want to calculate the accumulation function of the random variable FX (X ). For that, it is worth thinking about what the event {FX(X) ≤ x} has to do with the event {X ≤ Fx−1(x)}, and how this relates to the accumulation function.

Let X be some continuous random number whose density function is continuous and everywhere positive: fx(x) > 0 for all x ∈ R. Let FX be the accumulation function of this distribution. Show that the distribution of the transformation FX(X) is a continuous uniform distribution on the interval (0,1). Hint: You might want to calculate the accumulation function of the random variable FX (X ). For that, it is worth thinking about what the event {FX(X) ≤ x} has to do with the event {X ≤ Fx−1(x)}, and how this relates to the accumulation function.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

A continuous random variable T (in years) of the lifespan of a device has the following probability density function f (t) = {3 pt", Osts2 otherwise where p is a constant, (a) Determine the value of p. (b) Find the cumulative distribution function of T, F(t). (c) If U =ÉT+1 determine E(U) and Var(U).
Find Var(X) if X is a Uniformly distributed b Binomially distributed c Normally distributed By Dr. S. O. Edeki Department of Mathematics, Covenant University, Nigeria.
X is a random variable whose cumulative distribution function F(x) satisfies F(1) = 13, F(2) = 34, and F(3) = 1 • Make X discrete and give its probability mass function.• Make X continuous and give its probability density function.
Let X be a discrete random variable taking values {x1, x2, . . . , xn} with probability {p1, p2, . . . , pn}. The entropyof the random variable is defined asH(X) = −Σpilog (pi) Find the probability mass function for the above discrete random variable that maximizes the entropy.
Two discrete random variables X and Y have joint probability mass function (pmf) (a) (b) (c) ƒ(x) = { 5 0 Calculate the value of k. Show that f(x|y) Show that f(y x) = k n(n+1) = 1 n 1 8 x = 1,2,..., n; y=1,2,...,x. otherwise
Determine the probability density function of the random variable X* if x 1 ON|30
X1, X2, X3, and X4 are normally distributed random variables: X, - N(0,0), X, - N(0,1), X3 - N(1,0), and X4 - N(1,1). The variable(s) which follow(s) a standard normal distribution is (are) X, Y1, Y,, and Y, are normally distributed random variables. Use the normal cumulative distribution function to answer the following questions. If Y, - N(5,9), Pr (Y, 55.5) =. If Y2 - N(0,25), Pr (Y2 > 10) = If Y3 - N(29,16), Pr (20s Y3 5 40) =U. (Round your answers to four decimal places.)
Which of the following statements are true (no need to justify)?(1) The transformation of a continuous random variable is always a continuous random variable.(2) The distribution of the transformation of a random variable is always the same as that of the original sat-distribution of the univariate.(3) Many random variables can be represented by a uniformly distributed random variableusing if the quantile function of the distribution is known.(4) The accumulation function of several variables is increasing with respect to each variablefunction.(5) The multidimensional accumulation function is determined by its (one-dimensional) marginal distributionwhere from.(6) The marginal distributions of a discrete random vector are discrete distributions.