Chap 6-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 6 Continuous Random Variables and Probability Distributions Statistics for Business and Economics 6 th Edition

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-2 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, 6e © 2007 Pearson Education, Inc. Chap 6-3 Expectations for Continuous Random Variables  The mean of X, denoted μ X, is defined as the expected value of X  The variance of X, denoted σ X 2, is defined as the expectation of the squared deviation, (X - μ X ) 2, of a random variable from its mean

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-4 Linear Functions of Variables  Let W = a + bX, where X has mean μ X and variance σ X 2, and a and b are constants  Then the mean of W is  the variance is  the standard deviation of W is

Example 6.2: home heating costs  A homeowner estimates that within the range of likely temperatures her January heating bill, Y, in dollars, will be  Y = 290 – 5 T  Where T is the average temperature for the month, in degrees Fahrenheit.  If the average January temperature can be represented by a random variable with mean 24 and S.D. 4, find the mean and S.D. of this home owner’s January heating bill. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-5

Exercise 6.10  The profit for a production process is equal to $1000 minus 2 times the number of units produced. The mean and variance for the number of units produced are 50 and 90, respectively. Find the mean and variance of the profit. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-6

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-7  An important special case of the previous results is the standardized random variable  which has a mean 0 and variance 1 Linear Functions of Variables (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-8 The Normal Distribution  ‘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   Mean = Median = Mode x f(x) μ σ (continued)

Properties of Normal Distribution  Suppose that the Random variable X follows a normal distribution with parameters μ and σ 2. Then the following properties hold: 1.The mean of random variable is μ: E(X) = μ 2.The variance of random variable is σ 2 : Var(X) = E [(X-μ) 2 ] = σ 2 3.The shape of the probability density function is a symmetric bell-shaped curve centered on the mean (see Fig. 6.8) 4.We define the normal distribution using the notation X~N(μ,σ 2 ). Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-9

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-10 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 (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-11 By varying the parameters μ and σ, we obtain different normal distributions Many Normal Distributions

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-12 The Normal Distribution Shape x f(x) μ σ Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread. Given the mean μ and variance σ we define the normal distribution using the notation

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-13 Cumulative Normal Distribution  For a normal random variable X with mean μ and variance σ 2, i.e., X~N(μ, σ 2 ), the cumulative distribution function is x 0 x0x0 f(x)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-14 x bμa The probability for a range of values is measured by the area under the curve Finding Normal Probabilities

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-15 x bμa x bμa x bμa Finding Normal Probabilities (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-16 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 Z f(Z) 0 1

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-17 Example  If X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 is  This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100.

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-18 Comparing X and Z units Z X 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) (μ = 100, σ = 50) (μ = 0, σ = 1)

Example 6.4: normal probabilities Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-19 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 (will be given in the exam)

Example 6.3: 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, 6e © 2007 Pearson Education, Inc. Chap 6-20

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-21 f(X) X μ Probability as Area Under the Curve 0.5 The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-22 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 ) Z 0 a

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-23 The Standardized Normal Table Z Example: P(Z < 2.00) =.9772  Appendix Table 1 gives the probability F(a) for any value a

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-24 The Standardized Normal Table Z Example: P(Z < -2.00) = 1 – =  For negative Z-values, use the fact that the distribution is symmetric to find the needed probability: Z (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 6-25 General Procedure for Finding Probabilities  Draw the normal curve for the problem in terms of X  Translate X-values to Z-values  Use the Cumulative Normal Table To find P(a < X < b) when X is distributed normally: