Homework Questions

Transforming and Combining Random Variables Section 6.2 Transforming and Combining Random Variables

Jeep Tours Pete’s Jeep Tours offer a popular half-day trip to tourists. The number of passengers X on a randomly selected day has the following probability distribution. Find the expected number of passengers on any given day. Find the standard deviation: X 2 3 4 5 6 P(X) 0.15 0.25 0.35 0.20 0.05

Jeep Tours Pete charges $150 per passenger. The C=the total amount of money that Pete collects on a randomly selected trip. Find the mean and standard deviation X 2 3 4 5 6 C P(C)

Jeep Tours It costs Pete $100 to buy permits, gas, and a ferry pass for each half-day trip. The amount of profit $ that Pete makes from the trip is C-$100. Find the mean and standard deviation X 2 3 4 5 6 C $ P($)

Effects on Random Variables (we did this in chapter 2) Adding (or subtracting) a constant, a: Adds a to measures of center and location (mean, median, quartiles, percentiles) Does not change shape or measures of spread (range, IQR, standard deviation) Multiplying (or dividing) each observation by a constant 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

Effects of a Linear Transformation 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+𝑏 𝜇 𝑥 𝜎 𝑦 = 𝑏 𝜎 𝑥 (since b could be a negative number)

Combining 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 The mean of the sum of several random variables is the sum of their means.

Jeep Tours Pete’s sister, Erin, joins the company with her smaller car. What is her mean? (notation) How many total passengers, T, can Pete and Erin expect to have on their tours on a randomly selected day? Y 2 3 4 5 P(Y) 0.3 0.4 0.2 0.1

BUT… More variables means more variability!! When you add in another random variable you have to double check independence! Otherwise, you can’t calculate probabilities for T (total passengers)… Definition: If knowing whether any event involving X alone has occurred tells us nothing about the occurrence of any event involving Y alone, and vise versa, then X and Y are independent random variables.

Variance of the sum of independent random variables For any two independent random variables X and Y, if T = X + Y, then the variance of T is 𝜎 𝑇 2 = 𝜎 𝑥 2 + 𝜎 𝑌 2 In general, the variance of the sum of several independent random variables is the sum of their variances. Just remember – you can add variances only if the two random variables are independent, and that you CANNOT add standard deviations! (EVER)

Jeep Tours Find the standard deviation for T (total passengers):

Jeep Tours Erin charges $175 per passenger for her trip. Let G = the amount of money she collects on a randomly selected day. Find the mean and standard deviation of G. Let W = the total amount collected. Notice that W = C + G. Find the expected value and standard deviation (since number of passengers X and Y are independent)

Mean of the difference of 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 The mean of the sum of several random variables is the difference of their means.

Variance of the Difference of Random Variables For any two independent random variables X and Y, if D = X – Y , then the variance of D is 𝜎 𝐷 2 = 𝜎 𝑥 2 + 𝜎 𝑌 2

Combining Normal Random Variables If a random variable is Normally distributed, we can use its mean and standard deviation to compute probabilities.

Example – SUGAR! Mr. Starnes likes between 8.5 and 9 grams of sugar in his hot tea. Each morning, Mr. Starnes adds four randomly selected packets of sugar to his tea. Suppose the amount of sugar in each packet is Normally distributed with a mean of 2.17 grams and standard deviation 0.08 grams. What is the probability that Mr. Starnes tea tastes right?

Homework Pg 378 (35-39 odd, 40, 42, 44, 47, 48, 53, 54, 65, 66)