Download AP Statistics

AP Statistics
Chapter 16
Discrete Random Variables
A discrete random variable X has a countable number of
possible values. The probability distribution of X lists the
values and their probabilities.

Value of x
x1
x2
x3
……
xk
Probabilit
y
p1
p2
p3
……
pk


The probability pi must satisfy the rules of
probability.
We can use a probability histogram to
picture the probability distribution of a
discrete random variable.

The mean of a random variable x (x) is like
a weighted average that takes into account
that all the outcomes are not equally likely.
x is called the expected value. That does
not mean we expect to get that value on
any one trial, it means that we expect to
get that average value, overall, in the long
run.

Given a Discrete Probability Distribution:
Value of x
x1
x2
x3
……
xk
Probability
p1
p2
p3
……
pk





We do a weighted average to find the
mean
x = x1p1 + x2p2 + … + xkpk = xipi
The variance of X is
= (x1 - x)2p1 + (x2 - x)2p2 + … +
(xk - x)2pk = (xi - i)2pi


The Standard Deviation (x) of X is the
square root of the variance.
Both of the previous formulas are on the
AP Stat Formula Chart.

To evaluate the formulas we use the lists
since they make repetitive math easy.

E(x) = x = xipi can be evaluated by
storing the x’s in L1 and the probabilities in
L2. Let L3 be the products from both lists.
Now Sum L3.

Var (x) = (xi - i)2pi can also be
evaluated in the lists.
In L4, enter (L1 - x)2L2
remember x is the answer you got
when you summed L3
Now Sum L4
Rules for Means of Distributions:


1) If X is a random variable and you wish to
transform the data with an equation, you may
apply the same equation you would use on the
data to the old mean to create a new mean.
2) If X and Y are random variables and you wish
to combine or take the difference of their
distributions to create a new distribution, then you
either combine or take the difference
(respectively) of their means for a new
distribution.
Rules for Variances of
Distributions:



1) If X is a random variable and you wish to transform the
data with an equation, a) disregard any constant values in the
equation as they will not affect the variance b) to get the
variance of the new distribution, multiply the old variance by
any coefficient squared.
2) If X and Y are independent random variables and you
wish to create a new distribution by either adding or
subtracting the old distributions, then you may add (for sums
& differences) the old variances.
Remember: Variances of Random variables, add; standard
deviations do not.