Transcript Homework Questions

Homework Questions
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. X
2
3
4
5
P(X)
0.15
0.25
0.35
0.20
 Find the expected number of passengers on any
given day.
 Find the standard deviation:
6
0.05
Jeep Tours
 Pete charges $150 per passenger. The C=the total
amount of money that Pete collects on a randomly
selected trip.
X
2
3
4
5
C
P(C)
 Find the mean and standard deviation
6
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.
X
2
3
4
5
C
$
P($)
 Find the mean and standard deviation
6
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.
 𝜇𝑦 = 𝑎 + 𝑏𝜇𝑥
 𝜎𝑦 = 𝑏 𝜎𝑥 (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 = 𝜇𝑥 + 𝜇𝑦
 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.
Y
2
3
4
5
P(Y)
0.3
0.4
0.2
0.1
 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?
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
𝜎𝑇 2 = 𝜎𝑥 2 + 𝜎𝑌 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 = 𝜇𝑥 − 𝜇𝑦
 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
𝜎𝐷 2 = 𝜎𝑥 2 + 𝜎𝑌 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)