11 Chapter 4 Wrap Up
Concept Check
Section Reviews
4.1 Introduction
The characteristics of a probability distribution function (PDF) for a discrete random variable are as follows:
- Each probability is between zero and one, inclusive (inclusive means to include zero and one).
- The sum of the probabilities is one.
4.2 Measures of General DRVs
The expected value, or mean, of a discrete random variable predicts the long-term results of a statistical experiment that has been repeated many times. The standard deviation of a probability distribution is used to measure the variability of possible outcomes.
Mean or Expected Value: [latex]\mu = \underset{x\in X}{{\sum }^{\text{}}}xP\left(x\right)[/latex]
Standard Deviation: [latex]\sigma = \sqrt{\underset{x\in X}{{\sum }^{\text{}}}{\left(x-\mu \right)}^{2}P\left(x\right)}[/latex]
4.3 The Binomial Distribution
A statistical experiment can be classified as a binomial experiment if the following conditions are met:
- There are a fixed number of trials, n.
- There are only two possible outcomes, called “success” and, “failure” for each trial. The letter p denotes the probability of a success on one trial and q denotes the probability of a failure on one trial.
- The n trials are independent and are repeated using identical conditions.
The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials. The mean of X can be calculated using the formula μ = np, and the standard deviation is given by the formula σ = [latex]\sqrt{npq}[/latex].
X ~ B(n, p) means that the discrete random variable X has a binomial probability distribution with n trials and probability of success p.
X = the number of successes in n independent trials
n = the number of independent trials
X takes on the values x = 0, 1, 2, 3, …, n
p = the probability of a success for any trial
q = the probability of a failure for any trial
p + q = 1
q = 1 – p
The mean of X is μ = np. The standard deviation of X is σ = [latex]\sqrt{npq}[/latex].
Key Terms
Try to define the terms below on your own. Scroll over any term to check your response!
4.1 Introduction
- Random variable
- Probability model
- Discrete random variable
- Continuous random variable
- Probability mass function (PMF)
- Cumulative distribution function (CDF)
4.2 Measures of General DRVs
4.3 The Binomial Distribution
- Discrete random variable
- Binomial experiment
- Independent
- Bernoulli trial
- Probability mass function
- Cumulative distribution function
Extra Practice
4.1 Introduction
1. 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 the figure below using the data provided.
| x | P(x) |
|---|---|
| 0 |