Transcript Chapter 7 Section 7.2 Part 2

CHAPTER 7
Section 7.2 Part 2 – Means and Variances of
Random Variables
EFFECTS OF LINEAR TRANSFORMATION
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Recall from chapter 1 the effect of a linear
transformation on measures of center (mean &
median) and spread (standard deviation & IQR):
xnew = a + bx
Adding the same number, a, to each observation in a
data set adds a to measures of center and to quartiles
but does not change measures of spread.
Multiplying each observation in a data set by a
positive number, b, multiplies both measures of
center and measures of spread by b.
RULES FOR MEANS
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The rules for means and variances when working
with random variables are similar.
Rules for means:
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Rule 1: If X is a random variable and a and b are
fixed numbers, then
𝜇𝑎+𝑏𝑋 = 𝑎 + 𝑏𝜇𝑋
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Rule 2: If X and Y are random variables, then
𝜇𝑋+𝑌 = 𝜇𝑋 + 𝜇𝑌
See example 7.10 on p.419
RULES FOR VARIANCES
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Adding a constant value, a, to a random variable
does not change the variance, because the mean
increases by the same amount.
 2 a  X   X2
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Multiplying a random variable by a constant, b,
increases the variance by the square of the
constant.
 2bX  b 2 X2
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These two rules combined state the following:
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Rule 1: If X is a random variable and a and b are
fixed numbers, then
 2 a bX  b 2 X2
RULES FOR VARIANCES
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Two random variables X and Y are independent if the value of X has no
effect on the value of Y
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Rule 2: If X and Y are independent random variables, then
 2 X Y   X2   Y2
 2 X Y   X2   Y2
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Why add for the difference of variables?
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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.
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See example 7.11 on p.421-420
RULES FOR VARIANCES
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When variables are not independent, the variance of their sum
depends on the correlation between the two variables as well
as on their individual variances.
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For example, let X represent the amount of income spent and Y
represent the amount of income saved. When X increases, Y
decreases by the same amount. This relationship prevents their
variances from adding since their sum is always 100% and does
not vary at all.
The correlation between two random variables is represented by
𝜌, the Greek letter rho. This correlation coefficient has the same
properties as r.
Rule 3: If X and Y have correlation 𝜌, then
 2 X Y   X2   Y2  2  X  Y
 2 X Y   X2   Y2  2 X  Y
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This is the general addition rule for variances of random variables.
 If two variables are independent (not correlated), then 2  X  Y will
equal zero.
See example 7.12 on p.422-423
COMBINING NORMAL RANDOM VARIABLES
If a random variable is normally distributed, we
can use its mean and variance to compute
probabilities.
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following rule applies:
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Any linear combination of independent normal
random variables is also normally distributed.
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If X and Y are independent normal random variables and a
and b are any fixed numbers, aX+bY is also normally
distributed
See example 7.14 on p.424-425
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Homework: P.425 #’s 34, 36, 39, 44, 46, 49, & 50