Notes 6.2.notebook

Notes 6.2.notebook
December 14, 2011
6.2 Transforming and Combining Random Variables
Linear Transformations (remember effect of z­score calculations?)
Center/Locations
Spread
Shape
Add/Subtract a constant
Do same
No change No change
Multiply/Divide
by a constant
Do same
Do same
No change
Mean = 1.1 and standard deviation = 0.943.
1. Suppose the dealership's manager receives a $500 bonus for each car sold. Let Y = the bonus received from car sales during the first hour on a randomly selected Friday. Find the mean and standard deviation.
2. To encourage customers to buy cars on Friday mornings, the manager spends $75 to provide coffee and doughnuts. The manager's net profit T on a randomly selected Friday is the bonus earned minus this $75. Find the mean and std dev.
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Notes 6.2.notebook
December 14, 2011
Example: Scaling a Test
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?
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Notes 6.2.notebook
December 14, 2011
Combining Random Variables
If X and Y are INDEPENDENT random variables,
and T = X ± Y,
E(T) = µT = µX ± µY
Var(T) = σ2T = σ2X + σ2Y
You can NEVER add standard deviations!!! You MUST use the variances!
Hmm...Why don't we subtract variances when we do X ­ Y?
Why would we need to combine RVs?
Ex: Speed Dating
What is the probability that the man is taller than the woman in a randomly selected speed dating couple? We need to know about the distribution of men’s heights M, the distribution of women’s heights W, and what happens when we combine these distributions. Example: Speed Dating
(Continuous RVs)
Suppose that the height M of male speed daters has distribution N(70, 3.5) inches and the height F of female speed daters has distribution N(65, 3) inches. What is the probability that a randomly selected male speed dater is taller than the randomly selected female speed dater with whom he is paired? 3
Notes 6.2.notebook
December 14, 2011
Example: Apples
Suppose that the weights of 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?
(Discrete RVs)
Ex: Back to the Auto Dealership that is obsessed with Friday
morning sales...
X = # cars sold during first hour of business on a random Friday
Y = # cars leased during first hour of business on random Friday
0
Cars sold xi:
0.3
Probability pi:
Mean: μT = 1.1
3
2
1
0.1
0.2
0.4
Standard deviation: σT = 0.943
Cars leased yi:
0
Probability pi:
0.4
Mean: μT = 0.7
1
2
0.5
0.1
Standard deviation: σT = 0.64
1. Let T = total sales/leases = X + Y. Find and interpret μT and σT, assuming X and Y are independent.
2. The manager receives $500 bonus for each car sold and $300 for each car leased. Find the mean and standard deviation of B (bonus).
3. Find the probability that 1 car is sold and 2 are leased.
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