Download 6.2 Transforming and Combining Random

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.
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).
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.
What is the effect of multiplying or dividing a random variable by a constant?
Effect on a Random Variable of Multiplying (Dividing) 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.
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 fulltime 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.
What is the effect of adding (or subtracting) a constant to a random variable?
Effect on a Random Variable of Adding (or Subtracting) a Constant
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.
What is a linear transformation? How does a linear transformation
affect the mean and standard deviation of a random variable?
A linear transformation is a transformation of a random variable that involves adding a constant a,
multiplying by a constant b, or both.
Effect on a Linear Transformation on the Mean and Standard Deviation
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).
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.
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.
Rules for combining random variables
Many interesting statistics problems require us to examine two or more random variables.
Mean of the Sum of Random Variables
For any two random variables X and Y, if T = X + Y, then the expected value of T is
E(T) = µT = µX + µY
In general, the mean of the sum of several random variables is the sum of their means.
Variance 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
sT2 = sX2 + sY2
In general, the variance of the sum of several independent random variables is the sum of their
variances.
Rules for combining random variables
Mean 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
E(D) = µD = µX - µY
In general, the mean of the difference of several random variables is the difference of their means.
The order of subtraction is important!
Variance of the Difference of Random Variables
For any two random variables X and Y, if D = X - Y, then the variance of D is
sD2 = sX2 + sY2
In general, the variance of the difference of two independent random variables is the sum of their
variances.
• With regards to variance, you may be asking yourself “why add even for the
difference of the variables?” Think about this analogy:
We buy some cereal. The box says "16 ounces." We know that's not precisely the
weight of the cereal in the box, just close; after all, one corn flake more or less would
change the weight ever so slightly. Weights of such boxes of cereal vary somewhat,
and our uncertainty about the exact weight is expressed by the variance (or standard
deviation) of those weights.
Next we get out a bowl that holds 3 ounces of cereal and pour it full. Our pouring
skill certainly is not very precise, so the bowl now contains about 3 ounces with
some variability (uncertainty).
How much cereal is left in the box? Well, we'd assume about 13 ounces. But notice
that we're less certain about this remaining weight than we were about the weight
before we poured out the bowlful. The variability of the weight in the box has
increased even though we subtracted cereal.
Moral: Every time something happens at random, whether it adds to the pile or
subtracts from it, uncertainty (read "variance") increases.
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).
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 zscore 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.
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:
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.
The shape is also difficult to
determine.
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.