1. 6.2 Transforming and Combining Random Variables Objectives SWBAT: DESCRIBE the effects of transforming a random variable by adding or subtracting a constant and multiplying or dividing by a constant. FIND the mean and standard deviation of the sum or difference of independent random variables. FIND probabilities involving the sum or difference of independent Normal random variables.

2. El Dorado Community College considers a student to be full-time if he or she is taking between 12 and 18 units. The number of units X that a randomly selected El Dorado Community College full-time student is taking in the fall semester has the following distribution. Calculate and interpret the mean and standard deviation of X. If we were to randomly select many, many full-time students at El Dorado Community College, the average number of units they would be taking in the fall semester is 14.65. The number of units taken by randomly selected El Dorado Community College full-time students in the fall typically varies by about 2.06 units from the mean (14.65).

3. At El Dorado Community College, the tuition for full-time students is $50 per unit. So, if T = tuition charge for a randomly selected full-time student, T = 50X. Here’s the probability distribution for T: (note: 600=50(12 credits)) Calculate the mean and standard deviation of T.

4. What is the effect of multiplying or dividing a random variable by a constant? Multiplying (or dividing) each value of a random variable by a number b: Multiplies (divides) measures of center and location (mean, median, quartiles, percentiles) by b. Multiplies (divides) measures of spread (range, IQR, standard deviation) by |b|. Does not change the shape of the distribution. Effect on a Random Variable of Multiplying (Dividing) by a Constant

5. In addition to tuition charges, each full-time student at El Dorado Community College is assessed student fees of $100 per semester. If C = overall cost for a randomly selected full- time student, C = 100 + T. Here is the probability distribution for C and a histogram of the probability distribution: Calculate the mean and standard deviation of C.

6. What is the effect of adding (or subtracting) a constant to a random variable? Adding the same number a (which could be negative) to each value of a random variable: Adds a to measures of center and location (mean, median, quartiles, percentiles). Does not change measures of spread (range, IQR, standard deviation). Does not change the shape of the distribution. Effect on a Random Variable of Adding (or Subtracting) a Constant

7. What is a linear transformation? How does a linear transformation affect the mean and standard deviation of a random variable? If Y = a + bX is a linear transformation of the random variable X, then The probability distribution of Y has the same shape as the probability distribution of X. µ Y = a + bµ X. σ Y = |b|σ X (since b could be a negative number). Effect on a Linear Transformation on the Mean and Standard Deviation Linear transformations have similar effects on other measures of center or location (median, quartiles, percentiles) and spread (range, IQR). Whether we’re dealing with data or random variables, the effects of a linear transformation are the same. Note: These results apply to both discrete and continuous random variables. A linear transformation is a transformation of a random variable that involves adding a constant a, multiplying by a constant b, or both.

8. In a large introductory statistics class, the distribution of X = raw scores on a test was approximately normally distributed with a mean of 17.2 and a standard deviation of 3.8. The professor decides to scale the scores by multiplying the raw scores by 4 and adding 10. (a)Define the variable Y to be the scaled score of a randomly selected student from this class. Find the mean and standard deviation of Y. (b) What is the probability that a randomly selected student has a scaled test score of at least 90? N(78.8, 15.2) There is a 23.06% chance that a randomly selected student has a scaled test score of at least 90.

9. Rules for combining random variables Many interesting statistics problems require us to examine two or more random variables. For any two random variables X and Y, if T = X + Y, then the expected value of T is E(T) = µ T = µ X + µ Y In general, the mean of the sum of several random variables is the sum of their means. Mean of the Sum of Random Variables For any two independent random variables X and Y, if T = X + Y, then the variance of T is In general, the variance of the sum of several independent random variables is the sum of their variances. Variance of the Sum of Random Variables

10. Rules for combining random variables For any two random variables X and Y, if D = X - Y, then the expected value of D is E(D) = µ D = µ X - µ Y In general, the mean of the difference of several random variables is the difference of their means. The order of subtraction is important! Mean of the Difference of Random Variables For any two random variables X and Y, if D = X - Y, then the variance of D is In general, the variance of the difference of two independent random variables is the sum of their variances. Variance of the Difference of Random Variables

11. Speed Dating: Suppose that the height M of male speed daters follows a Normal distribution with a mean of 68.5 inches and a standard deviation of 4 inches and the height F of female speed daters follows a Normal distribution with a mean of 64 inches and a standard deviation of 3 inches. What is the probability that a randomly selected male speed dater is taller than the randomly selected female speed dater he is paired with? (only non-simulation approach) Let’s define the random variable D = M - F D will represent the difference between the male’s height and the female’s height. Because D is the difference of two independent Normal random variables, D follows a Normal distribution. Our goal is to find P(M > F), or P(D > 0).

12. We are working with a Normal distribution that has a mean difference of 4.5 inches and a standard deviation of 5 inches. If we were going to calculate this manually, we would substitute into our z- score formula using 0 for our x. We have to operate under the assumption that there is no difference between male and female heights. Using the standard Normal table, this gives us an area to the left of 0.1841. However, we are looking for the difference to be greater than 0 (M>F), so we need to find the area to the right, which is 1-0.1841 = 0.8159. On the calculator: There is a 0.8159 chance that a randomly selected male speed dater will be taller than the randomly selected female speed dater with whom he is paired.

13. Suppose that a certain variety of apples have weights that are approximately Normally distributed with a mean of 9 ounces and a standard deviation of 1.5 ounces. If bags of apples are filled by randomly selecting 12 apples, what is the probability that the sum of the 12 apples is less than 100 ounces? Let X = weight of a randomly selected apple Our goal: Find P(T<100) Because T is the sum of 12 independent Normal random variables, T follows a Normal distribution with:

14. N(108, 5.2) We want to find P(T<100) There is a 6.2% chance that the 12 randomly selected apples will have a total weight of less than 100 ounces.

15. The standard deviation cannot be calculated since the cost for tuition and fees and the cost for books are not independent. Students who take more units will typically have to buy more books. The shape is also difficult to determine.