Random Transformation 103

Random Transformation 103 14.1 method of distribution functions. one method that is often applicable is to compute the cdf of the transformed random variable, and if required, take the derivative to find the pdf. example let \ (x\) be a random variable with pdf given by \ (f (x) = 2x\), \ (0 \le x \le 1\). find the pdf of \ (y = 2x\). Simple addition of independent real valued random variables is perhaps the most important of all transformations. indeed, much of classical probability theory is concerned with sums of independent variables, in particular the law of large numbers , and the central limit theorem .

Random Transformation 103 Example 3. let xbe a uniform random variable on f n; n 1;:::;n 1;ng. then y = jxjhas mass function f y(y) = ˆ 1 2n 1 if x= 0; 2 2n 1 if x6= 0 : 2 continuous random variable the easiest case for transformations of continuous random variables is the case of gone to one. we rst consider the case of gincreasing on the range of the random variable x. Suppose that x and y are random variables on a probability space, taking values in r ⊆ r and s ⊆ r, respectively, so that (x, y) takes values in a subset of r × s. our goal is to find the distribution of z = x y. note that z takes values in t = {z ∈ r: z = x y for some x ∈ r, y ∈ s}. 14.1 introduction. in this section we will consider transformations of random variables. transformations are useful for: simulating random variables. for example, computers can generate pseudo random numbers which represent draws from \(u(0,1)\) distribution and transformations enable us to generate random samples from a wide range of more general (and exciting) probability distributions. 4.5. transformations of random variables. a function of a random variable is a random variable: if x x is a random variable and g g is a function then y = g(x) y = g ( x) is a random variable. in general, the distribution of g(x) g ( x) will have a different shape than the distribution of x x. the exception is when g g is a linear rescaling.

Random Transformation 103 14.1 introduction. in this section we will consider transformations of random variables. transformations are useful for: simulating random variables. for example, computers can generate pseudo random numbers which represent draws from \(u(0,1)\) distribution and transformations enable us to generate random samples from a wide range of more general (and exciting) probability distributions. 4.5. transformations of random variables. a function of a random variable is a random variable: if x x is a random variable and g g is a function then y = g(x) y = g ( x) is a random variable. in general, the distribution of g(x) g ( x) will have a different shape than the distribution of x x. the exception is when g g is a linear rescaling. Proposition. (transformation of discrete random variables) for each discrete random vector with joint pmf , the corresponding joint pmf of the transformed random vector where is bijective is. proof. considering the original pmf , we have in particular, the inverse exists since is bijective. A linear transformation is a change to a variable characterized by one or more of the following operations: adding a constant to the variable, subtracting a constant from the variable, multiplying the variable by a constant, and or dividing the variable by a constant. when a linear transformation is applied to a random variable, a new random.