A probability distribution function
is some function that may be used
to define a particular probability distribution.
A discreet probability distribution function
has two characteristics.
Number one, each probability is between
zero and one, inclusive,
meaning the probabilities do include zero and one.
And number two,
the sum of all the probabilities is equal to one.
As an example,
a child psychologist is interested
in the number of times a newborn baby's crying
wakes his mother after midnight.
So 50 random mothers were asked
how many times their newborn baby's crying, woke them
after midnight per week.
The following information was obtained.
First we let the random variable X
equal the number of times per week
a newborn's baby's crying wakes its mother after midnight.
For this example X is equal to
zero, one, two, three, four, and five.
This is a number of times the newborn babies crying
woke the mother after midnight in one week.
And P of X is equal to the probability
that the random variable X takes on the value of little X.
So again, the random variable X takes
on the values of zero one, two, three, four, and five
shown here in the first column,
looking at the table more closely,
the probability that X is equal to zero,
meaning the newborn baby's crying,
never woke the mother up in the week is equal to 2/50
the probability that X is equal to one is equal to 11/50.
The probability that X is equal to two is equal to 23/50
and so on.
This is a discreet probability distribution function because
each probability is between zero and one,
including zero and one.
And because the sum of the probabilities is equal to one.
I do want to mention a couple of things about capital X,
the random variable and little X.
The random variable is capital X and takes on the values,
small X.
So the connection between the values and the events
is that capital X equals small X
is the event that capital X takes the value of small X.
Let's take a look at another example.
Suppose Nancy has classes three times a week.
She attends classes three days a week, 80% of the time,
two days, 15% of the time, one day, four percent of the time
and zero days, one percent of the time.
Suppose one week is randomly selected.
For part A,
we let capital X the random variable
equal the number of days Nancy attends class,
capital X takes on the values of,
well she attends class three days, two days,
one day or zero days.
And therefore capital X takes on the values of
zero,
one,
two
and three.
And then part C suppose one day is randomly chosen
construct a probability distribution table
shown here on the right.
So the possible values of X are zero one, two and three.
The probably that X is equal to zero
would be the probability
she attends class zero days in the week,
which is one percent when normally expressed the probability
as a fraction or decimal,
I'm going to go ahead and give P of X
or P of zero as a decimal,
one percent is equal to 0.01.
Next we have the probability, that X is equal to one,
which should be the probability
she attends class one day in the week,
which is four percent or is it decimal 0.04.
Next we have the probability X is equal to two,
which is a probability she attends class two days,
which is 15% or as a decimal 0.15.
And then finally we have X equals three.
The probability X equals three
is the probably she attends class three days in the week,
which is 80% or is a decimal 0.8.
So they took additions
for discreet probability distribution function
is all the probabilities are between zero and one,
including zero and one, which is true.
And also the probabilities must sum to one.
So let's go and check the sum of these probabilities.
We have 0.01 plus 0.04 plus 0.15
plus 0.8, which is equal to one.
So this does confirm
this does represent a discreet probability
distribution function.
The content from this lesson
is from the OpenStax Introductory Statistics textbook,
which can be downloaded free
using the link shown here on the screen.
Thank you for watching.