Chapter 5 Continuous Random Variables and Probability Distributions Statistics for Business and Economics 7th Edition Chapter 5 Continuous Random Variables and Probability Distributions Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Continuous Probability Distributions A continuous random variable is a variable that can assume any value in an interval thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately. Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Expectations for Continuous Random Variables The mean of X, denoted μX , is defined as the expected value of X The variance of X, denoted σX2 , is defined as the expectation of the squared deviation, (X - μX)2, of a random variable from its mean Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Linear Functions of Variables (continued) An important special case of the previous results is the standardized random variable which has a mean 0 and variance 1 Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

The Normal Distribution (continued) ‘Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   f(x) σ x μ Mean = Median = Mode Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Properties of Normal Distribution Suppose that the Random variable X follows a normal distribution with parameters μ and σ2. Then the following properties hold: The mean of random variable is μ: E(X) = μ The variance of random variable is σ2: Var(X) = E [(X-μ)2] = σ2 The shape of the probability density function is a symmetric bell-shaped curve centered on the mean (see Fig. 5.8) We define the normal distribution using the notation X~N(μ,σ2). Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

The Normal Distribution (continued) 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 Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Many Normal Distributions By varying the parameters μ and σ, we obtain different normal distributions Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

The Normal Distribution Shape f(x) Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread. σ μ x Given the mean μ and variance σ we define the normal distribution using the notation Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Cumulative Normal Distribution For a normal random variable X with mean μ and variance σ2 , i.e., X~N(μ, σ2), the cumulative distribution function is f(x) x0 x Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Finding Normal Probabilities The probability for a range of values is measured by the area under the curve x a μ b Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Finding Normal Probabilities (continued) x a μ b x a μ b x a μ b Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

The Standardized Normal Any normal distribution (with any mean and variance combination) can be transformed into the standardized normal distribution (Z), with mean 0 and variance 1 Need to transform X units into Z units by subtracting the mean of X and dividing by its standard deviation f(Z) 1 Z Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Example If X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 is Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Comparing X and Z units 100 200 X 2.0 Z (μ = 100, σ = 50) 2.0 Z (μ = 0, σ = 1) Note that the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z) Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Example 5.4: normal probabilities If X~N(15,16), find the probability that X is larger than 18. Show relevant diagram What will be the probability that X is less than 18? Use Table 1 given in the Appendix Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Example 5.3: investment portfolio value A client has an investment portfolio whose mean value is equal to $500,000 with a S.D. of $15,000. She has asked you to determine the probability that the value of her portfolio is between $485,000 and $530,000. Also draw the normal distribution diagram for this problem. Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Probability as Area Under the Curve The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below f(X) 0.5 0.5 μ X Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

Appendix Table 1 The Standardized Normal table in the textbook (Appendix Table 1) shows values of the cumulative normal distribution function For a given Z-value a , the table shows F(a) (the area under the curve from negative infinity to a ) a Z Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

The Standardized Normal Table Appendix Table 1 gives the probability F(a) for any value a .9772 Example: P(Z < 2.00) = .9772 Z 2.00 Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

The Standardized Normal Table (continued) For negative Z-values, use the fact that the distribution is symmetric to find the needed probability: .9772 .0228 Example: P(Z < -2.00) = 1 – 0.9772 = 0.0228 Z 2.00 .9772 .0228 Z -2.00 Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

General Procedure for Finding Probabilities To find P(a < X < b) when X is distributed normally: Draw the normal curve for the problem in terms of X Translate X-values to Z-values Use the Cumulative Normal Table Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.