Continuous Random Variables and Continuous Distributions Continuous Random Variables and Continuous Distributions Expectation & Variance of Continuous Random Variables ( 5.2) The Uniform Random Variable ( 5.3) The Normal Random Variable ( 5.4) The Exponential Random Variable ( 5.5) The Gamma Random Variable ( 5.6.1) The Beta Random Variable ( 5.6.4) Transformation - Distributions of a Function of a Random Variable ( 5.7)
Continuous Random Variables and Distributions Expectation & Variance of Continuous Random Variables Expectation & Variance of Continuous Random Variables Definition X is said to be a continuous random variable if there exists a nonnegative function f, defined for all real values x (, ), having the property that for any set of real numbers P{X B} = f (x)dx. B Note 4.1: The function f is called the probability density function (p.d.f.) of the random variable X. Note 4.2: A function f is a probability density function if it satisfies the conditions 1. f (x) 0 for every x R. 2. f (x)dx = 1.
Continuous Random Variables and Distributions Expectation & Variance of Continuous Random Variables Expectation & Variance of Continuous Random Variables Example 4.1 If f (x) = 3 4 (1 x 2 ) for 1 x 1, and 0 otherwise. Is f (x) a probability density function? Solution: Note 4.3: P{a X b} = b a f (x)dx. Example 4.2 Using the probability density function in Example 4.2, find P { 0 X 1 2 } and P{0 X 3}. Solution:
Continuous Random Variables and Distributions Expectation & Variance of Continuous Random Variables Expectation & Variance of Continuous Random Variables Note 4.4: The probability that a continuous random variable will assume any fixed value is zero, that is a P{X = a} = f (x)dx = 0. a Example 4.3 Using the probability density function in Example 4.2, find P { X = 1 2 }. Solution:
Continuous Random Variables and Distributions Expectation & Variance of Continuous Random Variables Expectation & Variance of Continuous Random Variables Definition Given a continuous random variable X with probability density function f (x) for every x R, the cumulative distribution function of X is defined to be x F (x) = P{X x} = f (t)dt. Example 4.3 (Cont.) Note 4.5: The probability density function of a continuous random variable X can be found by differentiating the cumulative distribution function, such as That is f (x) = d dx F(x). F(x + δ) F(x) f (x) = lim = Slope of F(x) at point x. δ 0 δ
Continuous Random Variables and Distributions Expectation & Variance of Continuous Random Variables Expectation & Variance of Continuous Random Variables Example 4.3 (Cont.) Note 4.6: Intuitive interpretation of the density function f (a) The density f (a) is a measure of how likely the random variable X closed to a. Note 4.7: For any real values a < b, we have P{a X b} = F(b) F (a).
Continuous Random Variables and Distributions Expectation & Variance of Continuous Random Variables Expectation & Variance of Continuous Random Variables Expected value of a continuous random variable X Given a continuous random variable X with probability density function f (x), the Expect Value (or Mean) of X is defined to be Example 4.3 (Cont.) Find E(X). Solution: E(X) = xf (x)dx. x X
Continuous Random Variables and Distributions Expectation & Variance of Continuous Random Variables Expectation & Variance of Continuous Random Variables Proposition 4.1 Given a continuous random variable X with probability density function f (x), and let g be any real-valued function, then E[g(X)] = g(x)f (x)dx. x R Example 4.4 Using the probability density function in Example 4.2, and let g(x) = X 2, find E[g(X)]. Solution:
Continuous Random Variables and Distributions Expectation & Variance of Continuous Random Variables Expectation & Variance of Continuous Random Variables Corollary 4.1 Given a continuous random variable X with probability density function f (x), then E[aX + b] = ae(x) + b, where a and b are constants. Example 4.5 Using the probability density function in Example 4.2, and let g(x) = 5X + 8, find E[g(X)]. Solution:
Continuous Random Variables and Distributions Expectation & Variance of Continuous Random Variables Expectation & Variance of Continuous Random Variables Variance of a continuous random variable X Given a continuous random variable X with probability density function f (x) and mean µ, the Variance of X is defined to be Var(X) = E[(X µ) 2 ]. Proposition 4.2 Given a continuous random variable X with probability density function f (x) and mean µ, then Var(X) = E(X 2 ) µ 2. Corollary 4.2 Given a continuous random variable X with probability density function f (x), then where a and b are constants. Var(aX + b) = a 2 Var(X),
Continuous Random Variables and Distributions The Uniform Random Variable The Uniform Random Variable Definition A random variable X is said to be a uniform random variable on the interval (α,β) if and only if the probability density function of X is given by { 1 f (x) = β α, if α < x < β 0, otherwise, and denoted by X U(α,β). Proposition 4.3 If X U(α,β), then the cumulative distribution function of X is 0, x α x α F (x) = β α, α < x < β 1, x β.
Continuous Random Variables and Distributions The Uniform Random Variable The Uniform Random Variable Corollary 4.3 If X U(α,β), then E(X) = α + β 2 and Var(X) = (β α)2. 12
Continuous Random Variables and Distributions The Uniform Random Variable The Uniform Random Variable Note 4.7: If X U(0,1), then X has probability density function { 1, 0 < x < 1 f (x) = 0, otherwise. Note 4.8: If X U(0, 1), then the cumulative distribution function of X is 0, x 0 F (x) = x, 0 < x < 1. 1, x 1 Note 4.9: If X U(0,1), then E(X) = 1 2 and Var(X) = 1 12.
Continuous Random Variables and Distributions The Normal Random Variable The Normal Random Variable Definition A random variable X is said to be a normal random variable with parameters µ and σ 2 if and only if the probability density function of X is given by f (x) = 1 σ 2π exp and denoted by X N(µ,σ 2 ). { 1 2 ( x µ ) } 2, < x, µ <, σ > 0, σ We often refer the normal distribution as the Gaussian distribution. Note 4.10: The normal probability density function has a bell-shaped curve, symmetric about µ. Note 4.11: In particular, many random phenomena follow a normal probability distribution, at least approximately.
Continuous Random Variables and Distributions The Normal Random Variable The Normal Random Variable Note 4.12: If X N(µ,σ 2 ) with probability density function f (x), then f (x)dx = 1. Corollary 4.4 If X N(µ,σ 2 ), then E(X) = µ and Var(X) = σ 2.
Continuous Random Variables and Distributions The Normal Random Variable The Normal Random Variable Proposition 4.4 If X N(µ,σ 2 ), then Y = αx + β N(αµ + β,α 2 σ 2 ), where α > 0, < β <.
Continuous Random Variables and Distributions The Normal Random Variable The Normal Random Variable Note 4.13: If X N(µ,σ 2 ), then Z = X µ σ is said to have a standard normal distribution, and denoted by Z N(0, 1). Note 4.14: If Z N(0,1), then we traditionally denote its probability density function by φ(z), and its cumulative distribution function by Φ(z), that is z F (z) = P{Z z} = φ(x)dx.
Continuous Random Variables and Distributions The Normal Random Variable The Normal Random Variable Note 4.15: Table 5.1 (page 201) presents values of Φ(z) for non-negative z. Example 4.5 Find P{Z 1.96} from Table 5.1. Example 4.6 Find a such that P{Z a} = 0.95 from Table 5.1. Solution:
Continuous Random Variables and Distributions The Normal Random Variable The Normal Random Variable Note 4.16: For any x R, we have Φ( x) = 1 Φ(x). Example 4.7 Find P{Z 2.55} from Table 5.1. Note 4.17: If X N(µ,σ 2 ), then the cumulative distribution function of X can be written as ( ) x µ F (x) = Φ. σ
Continuous Random Variables and Distributions The Normal Random Variable The Normal Random Variable Example 4.8 If X N(5,4), Find P{4 X 7} from Table 5.1. Solution: Example 4.9 If X N(2,16), Find P{ X 1 < 3} from Table 5.1. Solution:
Continuous Random Variables and Distributions The Normal Random Variable The Normal Random Variable The Normal Approximation to the Binomial Distribution (The DeMoivre - Laplace Limit Theorem) If S n denotes the number of successes that occur when n independent trials, each resulting a success with probability p, are performed, then for any a < b P { a } S n np b Φ(b) Φ(a) as n. np(1 p) Note 4.18: Since S n Bin(n,p), we call S n np np(1 p) the standardized Bin(n,p) random variable, i.e., mean is 0 and variance is 1.
Continuous Random Variables and Distributions The Normal Random Variable The Normal Random Variable Note 4.19: There are two possible approximations to Binomial distribution. (1) Poisson approximation Good when n is large and np moderate. (2) Normal approximation Good when np(1 p) 10. Example 4.10 Given X Bin ( 100, 1 2 ), find P(48 X 52). Solution:
Continuous Random Variables and Distributions The Exponential Random Variable The Exponential Random Variable Definition A random variable X is said to be an exponential random variable with parameter λ > 0 if and only if the probability density function of X is given by and denoted by X Exp(λ). { λe f (x) = λx, ifx 0 0, otherwise, Note 4.20: In practice, the exponential random variable is often used to model the length of time until an event occurs.
Continuous Random Variables and Distributions The Exponential Random Variable The Exponential Random Variable Proposition 4.5 If X Exp(λ), then the cumulative distribution function of X is { 0, x < 0 F (x) = 1 e λx, x 0. Corollary 4.5 If X Exp(λ), then E(X) = 1 λ and Var(X) = 1 λ 2.
Continuous Random Variables and Distributions The Exponential Random Variable The Exponential Random Variable Definition A nonnegative random variable X is said to be memoryless if P{X > s + t X > t} = P{X > s}, for all s,t 0. Note 4.21: X Exp(λ) is a memoryless random variable, and is the only continuous memoryless random variable.
Continuous Random Variables and Distributions The Double Exponential Random Variable The Double Exponential Random Variable Definition A random variable X is said to have a double exponential (Laplace) distribution with parameter λ > 0 if and only if the probability density function of X is given by and denoted by X Laplace(λ). f (x) = 1 2 λe λ x, < x <. Proposition 4.6 If X Laplace(λ), then the cumulative distribution function of X is { 12 e F (x) = λx, x 0 1 1 2 e λx, x > 0.
Continuous Random Variables and Distributions The Double Exponential Random Variable The Double Exponential Random Variable Corollary 4.6 If X Laplace(λ), then E(X) = 0 and Var(X) = 2 λ 2.
Continuous Random Variables and Distributions The Gamma Random Variable The Gamma Random Variable Definition A random variable X is said to have a gamma distribution with parameters α > 0 and λ > 0 if and only if the probability density function of X is given by { λ α f (x) = Γ(α) x α 1 e λx, if x 0 0, otherwise, and denoted by X Gamma(α,λ), where Γ(α) = 0 t α 1 e t dt is called the gamma function. Note 4.22: For an integral value of n, we have Γ(n) = (n 1)!.
Continuous Random Variables and Distributions The Gamma Random Variable The Gamma Random Variable Note 4.23: When α = 1, the Gamma(α,λ) distribution is the Exp(λ) distribution. Note 4.24: In practice, the Gamma(α, λ) distribution with integer α > 0 often arise as the distribution of the amount of time until a total of α events has occurred. Note 4.25: If α = n 2 and λ = 1 2, (n > 0 integer), the Gamma(α,λ) distribution is called the chi-square distribution with n degree of freedom.
Continuous Random Variables and Distributions The Gamma Random Variable The Gamma Random Variable Corollary 4.7 If X Gamma(α,λ), then E(X) = α λ and Var(X) = α λ 2.
Continuous Random Variables and Distributions The Beta Random Variable The Beta Random Variable Definition A random variable X is said to have a beta distribution with parameters α > 0 and β > 0 if and only if the probability density function of X is given by f (x) = { Γ(α+β) Γ(α)Γ(β) x α 1 (1 x) β 1, if 0 < x < 1 0, otherwise, and denoted by X Beta(α,β). Note 4.26: When α = 1 = β, the Beta(α,β) random variable is the same as the U(0,1) random variable. Note 4.27: When α = β, the beta distribution is symmetric about x = 0.5.
Continuous Random Variables and Distributions The Beta Random Variable The Beta Random Variable Corollary 4.8 If X Beta(α,β), then E(X) = α α + β and Var(X) = αβ (α + β) 2 (α + β + 1).
Continuous Random Variables and Distributions Transformation Transformation Example 4.11 If X U(0,1), find the probability density function of Y = 2X + 1. Solution:
Continuous Random Variables and Distributions Transformation Transformation Theorem 4.1 Let X be a continuous random variable with probability density function f X (x). Suppose that g(x) is a strictly monotonic (increasing or decreasing), differential function of x, then the random variable g(x) has a probability density function given by { [ f f Y (y) = X g 1 (y) ] dy d g 1 (y), if y = g(x) for some x, 0, if y g(x) for all x where g 1 (y) is defined to equal that value of x such that g(x) = y. Example 4.12 Proof of Proposition 4.1 using Theorem 4.1