Continuous Random Variables

1. Continuous Random Variables • Let X be a random variable. Suppose there exists a nonnegative real-valued function f such that Then X is said to be a continuous random variable and f is the density function of X. Recall that the c.d.f. of X is • It follows that: (a) (b) (c) If f is continuous, F'(x) = f(x). (d)

2. Example of the density f and c.d.f. F for a random variable X y = f(x) y = F(x) This random variable X is called a uniform random variable over (0, 1). Also, X is the value of a random point selected from the interval (0, 1).

3. Continuous random variable—computing probability • Let X be a random variable with density function fX, where • Problems. Verify that fX is a density function. Find the probabilities: P(X = 0.5), P(0.5 < X < 0.6), P(0 < X < 0.5 or 0.6 < X < 1).

4. Function of a random variable • If we have a random variable X with a known density function fX, how can we get the density function for h(X), where h is a given function? • Suppose X has the density fX where Can you guess the density for h(X) = |X|?

5. Density function of a function of a random variable • We begin with an example showing how to get the density of the square of a random variable using the method of distribution functions. • If X has a continuous distribution function with probability density fX, then the distribution of Y = X2 is obtained by • Differentiation yields the density of Y:

6. The Method of Transformations • Theorem. Let X be a continuous random variable having density fX. Suppose that g is a strictly monotone, differentiable function of x. Then the random variable Y defined by Y = g(X) has density function • Example. Let X have density 2e–2x, x >0, and let g(x) = It follows from the previous theorem that Y = has density function

7. Expectations and Variances for Continuous Random Variables • If X is a continuous random variable with density f, the expected value of X is The expected value is also called the mean of X and it is also denoted as • We note that the expected value may fail to exist if the improper integral which defines it does not exist (as a finite value). We will assume that all expectations exist. • Theorem(LOUS). Let X be a continuous r. v. with density f. Then for any

8. Formulas for Mean and Variance • It follows from the definition of the mean and LOUS that • If X is a continuous r. v. with mean , then Var(X) and X, called the variance and standard deviation of X, resp., are • It can be shown that:

9. Example for mean and variance • Example. Let the density function be