Chapter 6: Discrete Probability Distributions

Notation Used in the Binomial Probability Distribution

1. There are n independent trials of the experiment. 2. Let p denote the probability of success so that 1 - p is the probability of failure. 3. Let X be a binomial random variable that denotes the number of successes in n independent trials of the experiment. So, 0 ≤ x ≤ n.

Trial

Each repetition of a probability experiment.

Discrete Random Variable

A ______________ random variable has either a finite or countable number of values. The values of a _________ random variable can be plotted as an isolated point on a number line.

Random variable (denoted by X)

A _____________variable is a numerical measure of the outcome from a probability experiment, so its value is determined by chance. _________ variables are typically denoted using letters such as X. When experiments are conducted in a way such that the outcome is a numerical​ result, it is said that the outcome is a random variable. A _________ variable is a numerical measure of the outcome of a probability​ experiment, so its value is determined by chance.

Probability histogram

The horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of each value of the random variable. The height of the rectangle is the probability rather than the frequency or relative frequency.

Interpretation of the Mean of a Discrete Random Variable

The mean of a discrete random variable can be thought of as the mean outcome of the probability experiment if we repeated the experiment many times. 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 the n trials will approach μX, the mean of the random variable X. In other words, let x-subcript 1 be the value of the random variable X after the first experiment, x-subscript 2 be the value of the random variable X after the second experiment, and so on. The difference between x-bar & μ-subscript x will get closer to 0 as n increases.

Binomial random variable

The number of successes in n trials of a binomial experiment.

Mean (or 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 and standard deviation given by the formulas Mean of X (the number of correct responses) = np e.g. n = 40 and p = .2 The square root of 40(.2)(1-.2) = 2.53

Poisson process

A random variable X, the number of successes in a fixed interval, follows a ___________ process provided the following conditions are met. The probability of two or more successes in any sufficiently small subinterval is 0. The probability of success is the same for any two intervals of equal length. The number of successes in any interval is independent of the number of successes in any other interval provided the intervals are not overlapping.

Continuous Random Variable

A random variable that has infinitely many values. These values can be plotted on a line in an uninterrupted fashion (no isolated points).

Binomial experiment

An experiment is said to be a ____________ 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. 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.

Expected value, E(X)

Because the mean of a random variable represents what we would expect to happen in the long run, it is also called the__________ value, E(X), of the random variable. Expected value (mean)

Probability distribution

The ___________ distribution of a discrete random variable X provides the possible values of the random variable and their corresponding probabilities. A__________ distribution can be in the form of a table, graph or mathematical formula. The probability of each value of the random variable is a number between 0 and 1. The probabilities over the entire distribution is always equal to 1.

Cumulative distribution function, P(X<=x)

Represents the probability of obtaining a random variable less than or equal to some specified value.

Rules for a Discrete Probability Distribution

Rules for ____________ probability distribution. Let P(x) denote the probability that the random variable X equals x; then 1. ΣP(x) = 1 2. 0 ≤ P(x) ≤ 1

The Mean of a Discrete Random Variable

The __________ of a discrete random variable is given by the formula... ...where x is the value of the random variable and P(x) is the probability of the observing the value x. The _________________ of a discrete random variable X is a weighted average of the possible values that the random variable can take. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi.

Binomial Probability Formula

The probability of obtaining x successes in n independent trials of a binomial experiment is given by...

Binomial Probability Distribution Function

The probability of obtaining x successes in n independent trials of a binomial experiment is given by:

Binomial probability distribution

The probability of obtaining x successes in n trials of a binomial experiment. The distribution becomes roughly symmetric when n is large.

Standard Deviation of a Discrete Random Variable

The standard deviation of a discrete random variable. A measure of spread for a distribution of a random variable that determines the degree to which the values differ from the expected value. The standard deviation of random variable X is often written as σ or σX. ... The square of the standard deviation is equal to the variance, Var(X) = σ2.

Notation Used in the Binomial Probability Distribution

There are n independent trials of the experiment. Let p denote the probability of success so that 1 − p is the probability of failure. Let X be a binomial random variable that denotes the number of successes in n independent trials of the experiment. So, 0 <(or equal to) x < (or equal to) n.