Chapter 16 Continuous Random Variables

1. Chapter 16 Continuous Random Variables

2. Chapter 16: Continuous Random Variables • Chapter Objectives • To introduce continuous random variables and discuss density functions. • To discuss the normal distribution, standard units, and the table of areas under the standard normal curve. • To show the technique of estimating the binomial distribution by using normal distribution.

3. Chapter 16: Continuous Random Variables • Chapter Outline 16.1) Continuous Random Variables The Normal Distribution The Normal Approximation to the Binomial Distribution 16.2) 16.3)

4. Chapter 16: Continuous Random Variables • 16.1 Continuous Random Variables • If X is a continuous random variable, the (probability) density function for X has the following properties:

5. Chapter 16: Continuous Random Variables • 16.1 Continuous Random Variables • Example 1 – Uniform Density Function The uniform density function over [a, b] for the random variable X is given by Find P(2 < X < 3). Solution: If [c, d] is any interval within [a, b] then For a = 1, b = 4, c = 2, and d = 3,

6. Chapter 16: Continuous Random Variables • 16.1 Continuous Random Variables • Example 3 – Exponential Density Function The exponential density function is defined by where k is a positive constant, called a parameter, whose value depends on the experiment under consideration. If X is a random variable with this density function, then X is said to have an exponential distribution. Let k = 1. Then f(x) = e−x for x ≥ 0, and f(x) = 0 for x < 0.

7. Chapter 16: Continuous Random Variables • 16.1 Continuous Random Variables • Example 3 – Exponential Density Function a. Find P(2 < X < 3). Solution: b. Find P(X > 4). Solution:

8. Chapter 16: Continuous Random Variables • 16.1 Continuous Random Variables • Example 5 – Finding the Mean and Standard Deviation If X is a random variable with density function given by find its mean and standard deviation. Solution:

9. Chapter 16: Continuous Random Variables • 16.2 The Normal Distribution • Continuous random variable X has a normal distribution if its density function is given by called the normal density function. • Continuous random variable Z has a standard normal distribution if its density function is given by called the standard normal density function.

10. Chapter 16: Continuous Random Variables • 16.2 The Normal Distribution • Example 1 – Analysis of Test Scores Let X be a random variable whose values are the scores obtained on a nationwide test given to high school seniors. X is normally distributed with mean 600 and standard deviation 90. Find the probability that X lies (a) within 600 and (b) between 330 and 870. Solution:

11. Chapter 16: Continuous Random Variables • 16.2 The Normal Distribution • Example 3 – Probabilities for Standard Normal Variable Z a. Find P(−2 < Z < −0.5). Solution: b.Find z0 such that P(−z0 < Z < z0) = 0.9642. Solution: The total area is 0.9642. By symmetry, the area between z = 0 and z = z0 is Appendix C shows that 0.4821 corresponds to a Z-value of 2.1.

12. Chapter 16: Continuous Random Variables • 16.3 The Normal Approximation to the • Binomial Distribution • Example 1 – Normal Approximation to a Binomial Distribution • The probability of x successes is given by Suppose X is a binomial random variable with n = 100 and p = 0.3. Estimate P(X = 40) by using the normal approximation. Solution: We have We use a normal distribution with

13. Chapter 16: Continuous Random Variables • 16.3 The Normal Approximation to the Binomial Distribution • Example 1 – Normal Approximation to a Binomial Distribution Solution (cont’d): Converting 39.5 and 40.5 to Z-values gives Therefore,