Understanding Mean and Standard Deviation of Discrete Random Variables, Study notes of Statistics

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Expected Values

The Expected Value of

X

Definition Let

X^ be a discrete rv with set of possible values

D^ and pmf

p ( x

). The

expected value

or^

mean value

of^

X , denoted by

E ( X

) or

μor just X^

μ, is

Example 16^ If we think of the population as consisting of the

X^ values 1,

2,... , 7, then

μ^ = 4.57 is the population mean.

In the sequel, we will often refer to

μ^ as the

population

mean

rather than the mean of

X^ in the population.

Notice that

μ^ here is not 4, the ordinary average of 1,... ,

7, because the distribution puts more weight on 4, 5, and 6than on other

X^ values.

cont’d

The Expected Value of a

Function

Example 23^ A computer store has purchased three computers of acertain type at $500 apiece. It will sell them for $1000apiece.^ The manufacturer has agreed to repurchase anycomputers still unsold after a specified period at $200apiece.^ Let

X^ denote the number of computers sold, and suppose that

p (0) = .1,

p (1) = .2,

p (2) = .3 and

p (3) = .4.

Example 23^ With

h ( X

) denoting the profit associated with selling X units, the given information implies that

h ( X

) = revenue – cost^ = 1000

X^ + 200(3 –

X ) – 1500

X^ – 900

The expected profit is then^ E

[ h ( X

)] =

h (0)

x^ p

h (1)

x^ p

h (2)

x^ p

h (3)

x^ p

cont’d

The Variance of

X

The Variance of

X

Definition Let

X^ have pmf

p ( x

) and expected value

μ. Then the

variance

of^

X , denoted by

V ( X

) or

2 σ X

, or just

2 σ, is

The

standard deviation

(SD) of

X^ is

The Variance of

X

So if

σ^ = 10, then in a long sequence of observed

X^ values,

some will deviate from

μ^ by more than 10 while others will

be closer to the mean than that—a typical deviation fromthe mean will be something on the order of 10.

Example 24^ A library has an upper limit of 6 on the number of videosthat can be checked out to an individual at one time.Consider only those who check out videos, and let

X

denote the number of videos checked out to a randomlyselected individual. The pmf of

X^ is as follows:

The expected value of

X^ is easily seen to be

μ^ = 2.85.

The Variance of

X

When the pmf

p ( x

) specifies a mathematical model for the

distribution of population values, both

2 σand

σ^ measure the

spread of values in the population;

2 σis the population

variance, and

σ^ is the population standard deviation.

A Shortcut Formula for

(^2) σ

The number of arithmetic operations necessary to compute^2 σcan be reduced by using an alternative formula. Proposition V ( X

-^ μ

2 =^

E ( X

2 ) – [

E ( X

2 )]

In using this formula,

E ( X

2 ) is computed first without any

subtraction; then

E ( X

) is computed, squared, and

subtracted (once) from

E ( X

Rules of Variance^ Proposition^ V (

aX^

+^ b

2 σ aX

=+ b

(^2) a

2 x σ

and x a

σ aX

+^ b^

In particular,σ aX

=^

,^ σ X

+^ b^

=^ σ

X

The absolute value is necessary because

a^ might be

negative, yet a standard deviation cannot be.Usually multiplication by

a^ corresponds to a change in the

unit of measurement (e.g., kg to lb or dollars to euros).

(3.14)

Rules of Variance^ According to the first relation in (3.14), the sd in the newunit is the original sd multiplied by the conversion factor.^ The second relation says that adding or subtracting aconstant does not impact variability; it just rigidly shifts thedistribution to the right or left.