Continuous Random Variables and Probability Distributions Chapter 6 Continuous Random Variables and Probability Distributions

Continuous Random Variables A random variable is continuous if it can take any value in an interval.

Cumulative Distribution Function The cumulative distribution function, F(x), for a continuous random variable X expresses the probability that X does not exceed the value of x, as a function of x

Shaded Area is the Probability That X is Between a and b a b x

Probability Density Function for a Uniform 0 to 1 Random Variable f(x) 1 1 x

Areas Under Continuous Probability Density Functions Let X be a continuous random variable with the probability density function f(x) and cumulative distribution F(x). Then the following properties hold: The total area under the curve f(x) = 1. The area under the curve f(x) to the left of x0 is F(x0), where x0 is any value that the random variable can take.

Properties of the Probability Density Function f(x) Comments Total area under the uniform probability density function is 1. 1 x0 1 x

Properties of the Probability Density Function Comments Area under the uniform probability density function to the left of x0 is F(x0), which is equal to x0 for this uniform distribution because f(x)=1. f(x) 1 x0 1 x

Reasons for Using the Normal Distribution The normal distribution closely approximates the probability distributions of a wide range of random variables. Distributions of sample means approach a normal distribution given a “large” sample size. Computations of probabilities are direct and elegant. The normal probability distribution has led to good business decisions for a number of applications.

Probability Density Function for a Normal Distribution 0.4 0.3 0.2 0.1 0.0  x

Probability Density Function of the Normal Distribution The probability density function for a normally distributed random variable X is Where  and 2 are any number such that - <  <  and - < 2 <  and where e and  are physical constants, e = 2.71828. . . and  = 3.14159. . .

Properties of the Normal Distribution Suppose that the random variable X follows a normal distribution with parameters  and 2. Then the following properties hold: The mean of the random variable is , The variance of the random variable is 2, The shape of the probability density function is a symmetric bell-shaped curve centered on the mean . By knowing the mean and variance we can define the normal distribution by using the notation

Effects of  on the Probability Density Function of a Normal Random Variable 0.4 Mean = 6 Mean = 5 0.3 0.2 0.1 0.0 x 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5

Effects of 2 on the Probability Density Function of a Normal Random Variable 0.4 Variance = 0.0625 0.3 0.2 Variance = 1 0.1 0.0 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 x

Cumulative Distribution Function of the Normal Distribution Suppose that X is a normal random variable with mean  and variance 2 ; that is X~N(, 2). Then the cumulative distribution function is This is the area under the normal probability density function to the left of x0,. As for any proper density function, the total area under the curve is 1; that is F() = 1.

Shaded Area is the Probability that X does not Exceed x0 for a Normal Random Variable f(x) x0 x

Range Probabilities for Normal Random Variables Let X be a normal random variable with cumulative distribution function F(x), and let a and b be two possible values of X, with a < b. Then The probability is the area under the corresponding probability density function between a and b.

Range Probabilities for Normal Random Variables f(x) a b x 

The Standard Normal Distribution Let Z be a normal random variable with mean 0 and variance 1; that is We say that Z follows the standard normal distribution. Denote the cumulative distribution function as F(z), and a and b as two numbers with a < b, then

Standard Normal Distribution with Probability for z = 1.25 0.8944 z 1.25

Finding Range Probabilities for Normally Distributed Random Variables Let X be a normally distributed random variable with mean  and variance 2. Then the random variable Z = (X - )/ has a standard normal distribution: Z ~ N(0, 1) It follows that if a and b are any numbers with a < b, then where Z is the standard normal random variable and F(z) denotes its cumulative distribution function.

Computing Normal Probabilities A very large group of students obtains test scores that are normally distributed with mean 60 and standard deviation 15. What proportion of the students obtained scores between 85 and 95? That is, 3.76% of the students obtained scores in the range 85 to 95.