Download Probability Distributions.

4.2 Probability Distributions
Definition. A random variable is a variable whose value is a numerical
outcome of a random phenomenon. The probability distribution of a
random variable tells us what the possible values of the variable are
and how probabilities are assigned to those values.
Discrete Random Variables
Definition. A discrete random variable X has a finite number of possible values. The probability distribution of X lists the values and their
probabilities:
Value of X x1 x2 x3 · · · xk
Probability p1 p2 p3 · · · pk
Example 4.9. A household is a group of people living together, regardless of their relationship to each other. Many sample surveys such
as the Current Population Survey select a random sample of households. Choose a household at random, and let the random variable X
be the number of people living there. Here is the distribution of X.
Household size
Probability
1
2
3
4
5
6
7
.251 .321 .171 .154 .067 .022 .014
The probability that a randomly chosen household has more than two
1
members is
P (X > 2) = P (X = 3) + P (X = 4) + P (X = 5) + P (X = 6) + P (X = 7)
= .171 + .154 + .067 + .022 + .014 = .428
Equally Likely Outcomes
Definition. If a random phenomenon has k possible outcomes, all
equally likely, then each individual outcome has probability 1/k. The
probability of any event A is
count of outcomes in A
count of all possible outcomes
count of outcomes in A
.
=
k
P (A) =
Example 4.10. Roll two dice and record the pips (dots) on each of the
two up-faces. Figure 4.8 (see TM-65) shows the 36 possible outcomes.
If the dice are carefully made, all 36 outcomes are equally likely. So
each has probability 1/36. Gamblers are often interested in the sum
of the pips on the up faces. What is the probability of rolling a 5?
The event “roll a 5” contains the four outcomes: (1,4), (2,3), (3,2),
(4,1). The probability is therefore 4/36 = 1/9 = 0.111. What about
the probability of rolling a 7? In Figure 4.8 (TM-65) you will find
six outcomes for which the sum of the pips is 7. The probability is
6/36 = 1/6 = 0.167.
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The Mean and Standard Deviation of a Discrete Random Variable
Definition. Suppose that X is a discrete random variable whose distribution is:
Value of X x1 x2 x3 · · · xk
Probability p1 p2 p3 · · · pk
Find the mean of X by multiplying each possible value by its probability
and adding over all the values:
µ = x 1 p1 + x 2 p2 + · · · + x k pk =
n
i=1
x i pi .
Note. The mean of a random variable X is a single fixed number µ.
It gives the average value of X in several senses:
• The mean µ is the average of the possible values of X, each
weighted by how likely it is to occur. That’s what the definition
of µ says.
• The mean µ is the point at which the probability histogram of
the distribution of X would balance if made of solid material. See
Figure 4.9 (and TM-66). Recall that the mean µ of a density curve
has this same property.
• If we actually repeat the random phenomenon many times, record
the value of X each time, and average these observed values, this
average will get closer and closer to µ as we make more and more
repititions. This fact is called the law of large numbers.
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Definition. Suppose that X is a discrete random variable whose distribution is:
Value of X x1 x2 x3 · · · xk
Probability p1 p2 p3 · · · pk
and that µ is the mean of X. The variance of X is
σ 2 = (x1 − µ)2 p1 + (x2 − µ)2 p2 + · · · + (xk − µ)2 pk =
n
(xi − µ)2pi .
i=1
The standard deviation σ is the square root of the variance.
Continuous Random Variables
Definition. A continuous random variable X takes all values in an
interval of numbers. The probability distribution of X is described by
a density curve. The probability of any event is the area under the
density curve and above the values of X that make up the event.
Note. The distribution of a continuous random variable assigns probabilities as areas under a density curve. See Figure 4.10 (and TM-67).
Definition (for those with some calculus background). Suppose
that X is a continuous random variable with probability distribution
P (X). The mean of X is
µ=
xP (x) dx
and the variance of X is
2
σ = (x − µ)2 P (x) dx,
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where the integrals are taken over all possible values of X. The standard
deviation σ is the square root of the variance.
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