Statistics 106 Ch. 6: Discrete Random Variables

Discrete Random Variable

Has either a finite or countable number of values. The values of a discrete random variable can be plotted on a number line with space between each point. Tend to stem from things that are counted.

Continuous Random Variable

Has infinitely many values. The values of a continuous random variable can be plotted on a number line in an uninterrupted fashion. These numbers normally stem from measurement (because there are multiple units of measurement).

Rules for a Discrete Probability Distribution

Let P(x) denote the probability that the random variable X equals x, then: 1. ∑P(x)=1 2. 0≤P(x)≤1

Binomial Random Variable

Let the random variable X be the number of successes in "n" trials of a binomial experiment. In this case, X is a binomial random variable.

Cumulative Distribution Function (CDF)

Used for computing probabilities less than or equal to a specified value. If trying to find an exact value, use PDF (binomial PDF).

Notation Used in the Binomial Probability Distribution

-There are "n" independent trials of the experiment. -Let "P" denote the probability of success for each trial so that 1-P is the probability of failure for each trial. -Let x denote the number of successes in "n" independent trials, so 0≤x≤n

Binomial Probability Graph Notes

For a fixed P, as the number of trials in a binomial experiment increases the probability distribution of the random variable X becomes bell shaped. As a rule of thumb, if n•p(1-P)≥10, the probability will be bell shaped. Because of the empirical rule, the interval µ-2(standard deviation) to µ+2(standard deviation) represents the usual observations.

The Mean of a Discrete Random Variable

µx=∑[x•P(x)] Where x is the value of the random variable and P(x) is the probability of observing the value x.

Mean (expected value) and Standard Deviation of a Binomial Random Variable

A binomial experiment with "n" independent trials and probability of success "p" has a mean given by the formula µx=n•p and a standard deviation given by the formula, standard deviation x=√(n•p(1-p))

Random Variable

A numerical measure of the outcome of a probability experiment, so its value is determined by chance. Random variables are typically denoted using capital letters such as "X" There are two types of random variables: discrete and continuous.

Criteria for a Binomial Probability Experiment

An experiment is said to be a binomial experiment if: 1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. 2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials. 3. For each trial there are two mutually exclusive (disjoint) outcomes: success or failure. 4. The probability of success is the same for each trial of the experiment.

Interpretation of the Mean of a Discrete Random Variable

Suppose an experiment is repeated "n" independent times and the value of the random variable X is recorded. As the number of repetitions of the experiment increases, the mean value of n trials will approach µx, the mean of the random variable X. In other words let x be the value of the random variable X after the first experiment, x2 be the value of the random variable X after the second experiment and so on. Then xbar= (x1+x2+x3...+xn)÷(n) The difference between xbar and µx gets closer to 0 as n increases.

Expected Value

The expected value of a discrete random variable, E(x), is the mean of the discrete random variable.

Probability Histogram

The horizontal axis corresponds to the value of the random variable X and the vertical axis represents the probability of each value of the random variable, so P(x).

Probability Distribution

The probability distribution of a discrete random variable x provides the possible values of the random variable and their corresponding probabilities. A probability distribution can be in the form of a table, graph or mathematical formula.

Binomial Probability Distribution Function

The probability of obtaining x successes in "n" independent trials of a binomial experiment is given by P(x)=nCx•(P^x)•(1-P)^(n-x) where "P" is the probability of success.

Standard Deviation of a Discrete Random Variable

standard deviation(x)= √∑[(x-µx)^2•P(x)] where x is the value of the random variable, µx is the mean of the random variable, and P(x) is the probability of observing a value of the random variable. The variance of a discrete random variable is the value under the square root in the computation of the standard deviation.