A discrete probability distribution function has two characteristics:
A child psychologist is interested in the number of times a newborn baby's crying wakes its mother after midnight. For a random sample of 50 mothers, the following information was obtained. Let X = the number of times per week a newborn baby's crying wakes its mother after midnight. For this example, x = 0, 1, 2, 3, 4, 5.
P(x) = probability that X takes on a value x.
x | P(x) |
---|---|
0 | P(x = 0) = |
1 | P(x = 1) = |
2 | P(x = 2) = |
3 | P(x = 3) = |
4 | P(x = 4) = |
5 | P(x = 5) = |
X takes on the values 0, 1, 2, 3, 4, 5. This is a discrete PDF because:
A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift. For a random sample of 50 patients, the following information was obtained. Let X = the number of times a patient rings the nurse during a 12-hour shift. For this exercise, x = 0, 1, 2, 3, 4, 5. P(x) = the probability that X takes on value x. Why is this a discrete probability distribution function (two reasons)?
X | P(x) |
---|---|
0 | P(x = 0) = |
1 | P(x = 1) = |
2 | P(x = 2) = |
3 | P(x = 3) = |
4 | P(x = 4) = |
5 | P(x = 5) = |
Each P(x) is between 0 and 1, inclusive, and the sum of the probabilities is 1, that is:
Suppose Nancy has classes three days a week. She attends classes three days a week 80% of the time, two days 15% of the time, one day 4% of the time, and no days 1% of the time. Suppose one week is randomly selected.
a. Let X = the number of days Nancy ____________________.
a. Let X = the number of days Nancy attends class per week.
b. X takes on what values?
b. 0, 1, 2, and 3
c. Suppose one week is randomly chosen. Construct a probability distribution table (called a PDF table) like the one in [link]. The table should have two columns labeled x and P(x). What does the P(x) column sum to?
c.
x | P(x) |
---|---|
0 | 0.01 |
1 | 0.04 |
2 | 0.15 |
3 | 0.80 |
Jeremiah has basketball practice two days a week. Ninety percent of the time, he attends both practices. Eight percent of the time, he attends one practice. Two percent of the time, he does not attend either practice. What is X and what values does it take on?
X is the number of days Jeremiah attends basketball practice per week. X takes on the values 0, 1, and 2.
The characteristics of a probability distribution function (PDF) for a discrete random variable are as follows:
Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. Over the years, they have established the following probability distribution.
Let X = the number of years a new hire will stay with the company.
Let P(x) = the probability that a new hire will stay with the company x years.
Complete [link] using the data provided.
x | P(x) |
---|---|
0 | 0.12 |
1 | 0.18 |
2 | 0.30 |
3 | 0.15 |
4 | |
5 | 0.10 |
6 | 0.05 |
x | P(x) |
---|---|
0 | 0.12 |
1 | 0.18 |
2 | 0.30 |
3 | 0.15 |
4 | 0.10 |
5 | 0.10 |
6 | 0.05 |
P(x = 4) = _______
P(x ≥ 5) = _______
0.10 + 0.05 = 0.15
On average, how long would you expect a new hire to stay with the company?
What does the column “P(x)” sum to?
1
x | P(x) |
---|---|
1 | 0.15 |
2 | 0.35 |
3 | 0.40 |
4 | 0.10 |
Define the random variable X.
What is the probability the baker will sell more than one batch? P(x > 1) = _______
0.35 + 0.40 + 0.10 = 0.85
What is the probability the baker will sell exactly one batch? P(x = 1) = _______
On average, how many batches should the baker make?
1(0.15) + 2(0.35) + 3(0.40) + 4(0.10) = 0.15 + 0.70 + 1.20 + 0.40 = 2.45
Define the random variable X.
Construct a probability distribution table for the data.
x | P(x) |
---|---|
0 | 0.03 |
1 | 0.04 |
2 | 0.08 |
3 | 0.85 |
We know that for a probability distribution function to be discrete, it must have two characteristics. One is that the sum of the probabilities is one. What is the other characteristic?
Define the random variable X.
Let X = the number of events Javier volunteers for each month.
What values does x take on?
Construct a PDF table.
x | P(x) |
---|---|
0 | 0.05 |
1 | 0.05 |
2 | 0.10 |
3 | 0.20 |
4 | 0.25 |
5 | 0.35 |
Find the probability that Javier volunteers for less than three events each month. P(x < 3) = _______
Find the probability that Javier volunteers for at least one event each month. P(x > 0) = _______
1 – 0.05 = 0.95
Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given in [link].
x | P(x) |
---|---|
3 | 0.05 |
4 | 0.40 |
5 | 0.30 |
6 | 0.15 |
7 | 0.10 |