Chapter 4: Discrete Random Variables
Chapter 4 Review
Section 4.1 Review
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.
Section 4.2 Review
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.
Formula Review
Mean or Expected Value: [latex]\mu =\sum\limits_{x\in X} xP\left(x\right)[/latex]
Standard Deviation: [latex]\sigma =\sqrt{\sum\limits_{x\in X}{\left(x-\mu \right)}^{2}P\left(x\right)}[/latex]
Section 4.3 Review
A statistical experiment can be classified as a binomial experiment if the following conditions are met:
- There are a fixed number of trials [latex]n[/latex].
- There are only two possible outcomes, called “success” and “failure” for each trial. The letter [latex]p[/latex] denotes the probability of a success on one trial and [latex]q[/latex] denotes the probability of a failure on one trial.
- The [latex]n[/latex] trials are independent and are repeated using identical conditions.
The outcomes of a binomial experiment fit a binomial probability distribution. The random variable [latex]X = \text{the number of successes obtained in the } n \text{ independent trials}[/latex]. The mean of [latex]X[/latex] can be calculated using the formula [latex]\mu = np[/latex], and the standard deviation is given by the formula [latex]\sigma = \sqrt{npq}[/latex].
Formula Review
[latex]X \sim B(n,p)[/latex] means that the discrete random variable [latex]X[/latex] has a binomial probability distribution with [latex]n[/latex] trials and probability of success [latex]p[/latex].
[latex]X= \text{the number of successes in } n \text{ independent trials}[/latex]
[latex]n = \text{the number of independent trials}[/latex]
[latex]X[/latex] takes on the values [latex]x = 0, 1, 2, 3, \ldots, n[/latex]
[latex]p= \text{the probability of a success for any trial}[/latex]
[latex]q = \text{the probability of a failure for any trial}[/latex]
[latex]p+q=1[/latex]
[latex]q = 1-p[/latex]
The mean of [latex]X[/latex] is [latex]\mu = np[/latex]. The standard deviation of [latex]X[/latex] is [latex]\sigma = \sqrt{npq}[/latex].
Section 4.4 Review
There are three characteristics of a geometric experiment:
- There are one or more Bernoulli trials with all failures except the last one, which is a success.
- In theory, the number of trials could go on forever. There must be at least one trial.
- The probability, [latex]p[/latex], of a success and the probability, [latex]q[/latex], of a failure are the same for each trial.
In a geometric experiment, define the discrete random variable [latex]X[/latex] as the number of independent trials until the first success. We say that [lat