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6.2 Transforming and Combining Random Variables

In Chapter 2, we studied the effects of transformations on the ___________________________________ of a distribution of data.

1. Adding (or Subtracting) a Constant Adding the same number A (either positive, zero, or negative) to each observation:

x Adds A to measures of center and location (_______________________________________________________). x Does not change shape or measures of spread (____________________________________________________).

2. Multiplying (or Dividing) by a Constant

Multiplying (or Dividing) each observation by a constant B (positive, negative, or zero): x Multiplies (divides) measure of center and location (mean, median, quartiles, percentiles) by B. x Multiplies (divides) measures of spread (range, IQR, standard deviation) by B. x Does not change the shape of the distribution.

Example 1: Pete’s Jeep Tours offers a popular half-day trip in a tourist area. There must be at least 2 passengers for the trip to run, and the vehicle will hold up to 6 passengers. The number of passengers X on a randomly selected day has the given probability distribution. Use technology to: a) Find the mean, standard deviation, and variance of X. b) Make a histogram of the distribution. Example 2: Pete charges $150 per passenger. Let C = the total amount of money that Pete collects on a randomly selected trip. Find the probability distribution of C. Use technology to: a) Find the mean, standard deviation, and variance of X. b) Make a histogram of the distribution.

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Example 3: It costs Pete $100 to buy permit, gas, and a ferry pass for each half-day trip. The amount of profit, V , that Pete makes from the trip is the total amount of money C that he collects from the passengers minus $100. Use technology to: a) Find the mean, standard deviation, and variance of X. b) Make a histogram of the distribution. In summary:

1. Shape: Neither transformation changes the shape of the probability distribution. 2. Center: The mean of X is multiplied by 150 and then decreased by 100. 3. Spread: The standard deviation of X is multiplied by 150 and is unchanged by the subtraction.

Example 4: Create an equation for the linear transformation V in example 4 above. Then… a) Find the mean of V. b) Find the standard deviation of V. Effects of a Linear Transformation on the Mean and Standard Deviation If Y a bX � is a linear transformation of the random variable X, then

1. The probability distribution of Y has the same shape as the probability distribution of X. 2. _______________

3. _______________ (since b could be a negative number)

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Example 5: One brand of bathtubs comes with a dial to set the water temperature. When the “babysafe” setting is selected and the tub is filled, the temperature X of the water follows a Normal distribution with a mean of 34°C and a standard deviation of 2°C. a) Define the random variable Y to be the water temperature in degrees Fahrenheit when the dial is set on “babysafe”

(recall that 9 5 32F C � ). Find the mean and standard deviation of Y. b) According to Babies R Us, the temperature of baby’s bathtub should be between 90°F and 100°F. Find the probability that the water temperature on a randomly selected day when the “babysafe” setting is used meets the Babies R Us recommendation. Show your work. Independent Random Variables – if knowing whether any event involving X alone has occurred tells us nothing about the occurrence of an event involving Y alone, and vice versa, then X and Y are ________________________________________. Mean of the Sum/Difference of Random Variables For any two random variables X and Y, if T = X + Y, then the expected value of T is

In general, the mean of the sum/difference of several random variables is the sum/difference of their means. Variance of the Sum/Difference of Independent Random Variables For any two _______________ random variables X and Y, if T = X + Y, then the variance of T is

In general, the variance of the sum/difference of several _______________ random variables is the sum of their variances. If the random variables are not independent then this Example 6: A college uses SAT scores as one criterion for admission. Experience has shown that the distribution of SAT scores among its entire population is such that:

SAT Math Score X: 519 115X XP V

SAT Critical Reading Score Y: 507 111Y YP V

SAT English Score Y: 522 104Z ZP V Find the overall average and standard deviation of SAT scores.

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