Continuous Random Variables Chapter Five Continuous Random Variables McGraw-Hill/Irwin Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

Continuous Random Variables 5.1 Continuous Probability Distributions 5.2 The Uniform Distribution 5.3 The Normal Probability Distribution *5.4 Approximating the Binomial Distribution by Using the Normal Distribution *5.5 The Exponential Distribution *5.6 The Cumulative Normal Table

5.1 Continuous Probability Distributions The curve f(x) is the continuous probability distribution (or probability curve or probability density function) of the random variable x if the probability that x will be in a specified interval of numbers is the area under the curve f(x) corresponding to the interval. Properties of f(x) 1. f(x)  0 for all x 2. The total area under the curve of f(x) is equal to 1

5.2 The Uniform Distribution If c and d are numbers on the real line, the probability curve describing the uniform distribution is The mean and standard deviation of a uniform random variable x are

The Uniform Probability Curve

5.3 The Normal Probability Distribution The normal probability distribution is defined by the equation and  are the mean and standard deviation,  = 3.14159 … and e = 2.71828 is the base of natural or Naperian logarithms.

The Position and Shape of the Normal Curve

Normal Probabilities

Three Important Areas under the Normal Curve The Empirical Rule for Normal Populations

The Standard Normal Distribution If x is normally distributed with mean  and standard deviation , then is normally distributed with mean 0 and standard deviation 1, a standard normal distribution.

Some Areas under the Standard Normal Curve

Calculating P(z  -1)

Calculating P(z  1)

Finding Normal Probabilities Example 5.2 The Car Mileage Case Procedure Formulate in terms of x. Restate in terms of relevant z values. 3. Find the indicated area under the standard normal curve.

Finding Z Points on a Standard Normal Curve

Finding X Points on a Normal Curve Example 5.5 Finding the number of tapes stocked so that P(x > st) = 0.05

Finding a Tolerance Interval Finding a tolerance interval [  k] that contains 99% of the measurements in a normal population.

5.4 Normal Approximation to the Binomial If x is binomial, n trials each with probability of success p and n and p are such that np  5 and n(1-p)  5, then x is approximately normal with

Example: Normal Approximation to Binomial Example 5.8: Approximating the binomial probability P(x = 23) by using the normal curve when

5.5 The Exponential Distribution If l is positive, then the exponential distribution is described by the probability density function mean mx=1/l standard deviation sx=1/l

Example: Computing Exponential Probabilities Given mx=3.0 or l=1/3=.333, 0.05 0.1 0.15 0.2 0.25 0.3 0.35 5 9 l=0.333 x mx

5.6 The Cumulative Normal Table The cumulative normal table gives of being less than or equal any given z-value The cumulative normal table gives the shaded area

Discrete Random Variables Summary: 5.1 Continuous Probability Distributions 5.2 The Uniform Distribution 5.3 The Normal Probability Distribution *5.4 Approximating the Binomial Distribution by Using the Normal Distribution *5.5 The Exponential Distribution *5.6 The Cumulative Normal Table