Download 6.2 Transforming and Combining Random Variables

6.2 Transforming and Combining Random Variables.notebook
December 17, 2014
6.2
Transforming and Combining
Random Variables
6.2A
Linear
Transformations
6.2 Transforming and Combining Random Variables.notebook
December 17, 2014
Example
El Dorado Community College considers a student to be full­time if he or she is taking between 12 and 18 credits. The number of credits X that randomly selected full­time student is taking in the fall semester has the following distribution:
X
12
13
14
15
16
17
18
P(X)
0.25
0.10
0.05
0.30
0.10
0.05
0.15
a) Calculate and interpret the mean and standard deviation X.
X
12
13
14
15
16
17
18
P(X)
0.25
0.10
0.05
0.30
0.10
0.05
0.15
The average number of credits taken by a randomly
selected El Dorado student is _______ credits.
On average, the number of credits taken by a randomly
selected El Dorado student differs from the mean by
________ credits.
6.2 Transforming and Combining Random Variables.notebook
December 17, 2014
b) Suppose that the tuition (T) for full­time students is $50 per credit. Find the mean and standard deviation of the tuition charges. What is the effect of multiplying or dividing a random variable by a constant (shape, center, spread)?
T
P(T)
0.25
0.10
0.05
0.30
0.10
0.05
0.15
c) In addition to tuition charges, each full­time student is assessed student fee of $100 per semester. Find the mean and standard deviation of total charges(C). What is the effect of adding (or subtracting) a constant to a random variable? X
P(X)
0.25
0.10
0.05
0.30
0.10
0.05
0.15
6.2 Transforming and Combining Random Variables.notebook
December 17, 2014
d) What linear equation can be constructed to transform from X to C?
Effects of Transformations on the Shape, Center, and Spread (from Chapter 2):
1. Adding or subtracting a constant:
a. Adds or subtracts the constant to measures of center and location (mean, median, quartiles, percentiles) b. Does not change shape or spread (range, IQR, standard deviation)
2. Multiplying or dividing by a constant
a. Multiplies or divides measures of center and location by the constant
b. Multiplies or divides measures of spread by absolute value of the constant
c. Does not change the shape
THE RULES FROM ABOVE ALSO APPLY TO DISCRETE AND RANDOM VARIABLES!!
6.2 Transforming and Combining Random Variables.notebook
December 17, 2014
Effects of a Linear Transformation on the Mean and Standard Deviation (discrete and continuous):
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.
Example
In a large 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.
6.2 Transforming and Combining Random Variables.notebook
December 17, 2014
b) What is the probability that a randomly selected student has a scaled test score of at least 90?
6.2B
Combining Random Variables,
Combining Normal Random
Variables
6.2 Transforming and Combining Random Variables.notebook
December 17, 2014
Mean and Variance of the Sum of Random Variables:
For any two variables X and Y, if T = X + Y, then the expected value of T is
For any two INDEPENDENT random variables X and Y, if T = X + Y, then the variance of T is
6.2 Transforming and Combining Random Variables.notebook
December 17, 2014
AP Exam Common Error: Many students lose credit when combining two or more random variables because they add the standard deviations instead of adding the variances.
Example
El Dorado Community College also has a campus downtown, specializing in just a few fields of study. Full­time students at the downtown campus take only 3 credit classes. Let Y = number of credits taken in the fall semester by a randomly selected full­time student at the downtown campus. Here is the probability distribution of Y:
Y
12
15
18
P(Y)
0.3
0.4
0.3
6.2 Transforming and Combining Random Variables.notebook
December 17, 2014
(a)If you were to randomly select one full­time student from the main campus and one full­time student from the downtown, what is the expected value of the sum of credits?
(b)What is the standard deviation of the sum of credits?
6.2 Transforming and Combining Random Variables.notebook
December 17, 2014
Example
Let B = the amount spent on books in the fall semester for a randomly selected full­time student at El Dorado Community College. Suppose that and . Recall from earlier that C = overall cost for tuition and fees for a randomly selected full­time student at the community college and that and
Find the mean, standard deviation, and shape of the cost of tuition, fees, and books (C + B) for a randomly selected full­time student.
Mean and Variance of the Difference of Random Variables:
For any two random variables X and Y, if D = X – Y, then the expected value of D is 6.2 Transforming and Combining Random Variables.notebook
December 17, 2014
For any two INDEPENDENT random variables X and Y, if D = X – Y, then the variance of D is
Example At the downtown campus, full­time students pay $55 per unit. Let U = cost of tuition for a randomly selected full­time student at the downtown campus. (a)Find the mean and standard deviation of U.
6.2 Transforming and Combining Random Variables.notebook
(b)Suppose we randomly select one full­time student from each of the two campuses. What are the mean and standard deviation of the difference in tuition charges, D = T – U? Interpret each value.
Example 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 weights of the 12 apples is less than 100 ounces?
December 17, 2014
6.2 Transforming and Combining Random Variables.notebook
Example Suppose that the height M of male speed daters follow a Normal distribution, with a mean of 70 inches and a standard deviation of 3.5 inches, and suppose that the height F of female speed daters follows a Normal distribution, with a mean of 65 inches and a standard deviation of 3 inches. What is the probability that a randomly selected male speed dater is taller than a randomly selected female speed dater with whom he is paired?
December 17, 2014