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 [latex]X[/latex] has a geometric distribution and write [latex]X \sim G(p)[/latex].
Formula Review
[latex]X \sim G(p)[/latex] means that the discrete random variable [latex]X[/latex] has a geometric probability distribution with probability of success in a single trial [latex]p[/latex].
[latex]X = \text{the number of independent trials until the first success}[/latex]
[latex]X[/latex] takes on the values [latex]x = 1, 2, 3, \ldots[/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 is [latex]\mu = \frac{1}{p}[/latex].
The standard deviation is [latex]\sigma = \sqrt{\frac{1–p}{p^2}} = \sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}[/latex].
Section 4.5 Review
A hypergeometric experiment is a statistical experiment with the following properties:
- You take samples from two groups.
- You are concerned with a group of interest, called the first group.
- You sample without replacement from the combined groups.
- Each pick is not independent, since sampling is without replacement.
- You are not dealing with Bernoulli Trials.
The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable [latex]X= \text{the number of items from the group of interest}[/latex]. The distribution of[latex]X[/latex] is denoted [latex]X \sim H(r,b,n)[/latex], where [latex]r= \text{the size of the group of interest (first group)}[/latex], [latex]b= \text{the size of the second group}[/latex], and [latex]n = \text{the size of the chosen sample}[/latex]. It follows that [latex]n \leq r+b[/latex]. The mean of [latex]X[/latex] is [latex]\mu = \frac{nr}{r+ b}[/latex] and the standard deviation is [latex]\sigma = \sqrt{\frac{rbn(r+b− n)}{(r + b)^{2}(r + b-1)}}[/latex].
Formula Review
[latex]X \sim H(r,b,n)[/latex] means that the discrete random variable [latex]X[/latex] has a hypergeometric probability distribution with [latex]r= \text{the size of the group of interest (first group)}[/latex], [latex]b= \text{the size of the second group}[/latex], and [latex]n = \text{the size of the chosen sample}[/latex].
[latex]X= \text{the number of items from the group of interest that are in the chosen sample}[/latex]
[latex]X[/latex] may take on the values [latex]x = 0, 1, 2, \ldots[/latex] up to the size of the group of interest. (The minimum value for [latex]X[/latex] may be larger than zero in some instances.)
[latex]n \leq r+b[/latex]
[latex]\mu = \frac{nr}{r+ b}[/latex]
[latex]\sigma = \sqrt{\frac{rbn(r+b− n)}{(r + b)^{2}(r + b-1)}}[/latex]
Section 4.6 Review
A Poisson probability distribution of a discrete random variable gives the probability of a number of events occurring in a fixed interval of time or space, if these events happen at a known average rate and independently of the time since the last event. The Poisson distribution may be used to approximate the binomial, if the probability of success is “small” (less than or equal to 0.05) and the number of trials is “large” (greater than or equal to 20).
Formula Review
[latex]X \sim P(\mu)[/latex] means that [latex]X[/latex] has a Poisson probability distribution where [latex]X= \text{the number of occurrences in the interval of interest}[/latex].
[latex]X[/latex] takes on the values [latex]x = 0, 1, 2, 3, \ldots[/latex]
The mean [latex]\mu[/latex] is typically given.
The variance is [latex]\sigma^2 = \mu[/latex], and the standard deviation is
[latex]\sigma \text{ = }\sqrt{\mu }[/latex].
When [latex]P(\mu)[/latex] is used to approximate a binomial distribution, [latex]\mu = np[/latex] where [latex]n[/latex] represents the number of independent trials and [latex]p[/latex] represents the probability of success in a single trial.
a number that measures how far the outcomes of a statistical experiment are from the mean of the distribution