Normal and Discrete Probability Distributions

This chapter covers normal and discrete probability distributions.

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.

discrete random variable

Has either a finite or countable number of values. The values can be plotted on a number line with space between each point.

continuous random variable

Has infinitely many values. The values can be plotted on a line in an uninterrupted fashion.

probability distribution

Provides the possible values of the random variable and their corresponding probabilities. Can be in the form of a table, graph, or mathematical formula.

rules for a discrete probability distribution

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

probability histogram

a histogram in which 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

mean of a discrete random variable

µ-sub-x = ∑ [x ∗ P(x)]

where x is the value of the random variable and P(x) is the probability of observing the variable x.

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 the n trials will approach µx, the mean of the random variable X.

x̄ =( x₁ + x₂ + ⋅⋅⋅ + x-sub-n)/n

The difference between x̄ and µ-sub-x gets closer to 0 as n increases

expected value

the mean of the discrete random variable

variance of a discrete random variable

σ²-sub-x = ∑ [x² P(x)] -µ²-sub-x

standard deviation of a discrete random variable

σ-sub-x = √σ²-sub-x

criteria for a binomial probability experiment

1. The experiment is performed a fixed number of times. Each repetition 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.

binomial random variable

the number of successes in n trials of a binomial experiment

notation used in binomial probability distribution

1. There are n independent trials of the experiment.
2. P denotes the probability of success for each trial so that 1-p is the probability of failure for each trial.
3. X denotes the number of successes in n independent trials of the experiment. 0 ≤ x ≤ 1.

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)

x = 0,1, 2, ⋅⋅⋅,n
where p is the probability of success

cumulative distribution function

computes probabilities less than or equal to a specified value

mean 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

u-sub-x = np and σ-sub-x = √np(1-p)

uniform probability distribution

All values have the same probability

probability density function

an equation used to compute probabilities of continuous random variables.

requirements of a probability density function

1. The total area under the graph of the equation over all possible values of the random variable must equal 1
2. The height of the graph of the equation must be greater than or equal to 0 for all possible value of the random variable. In other words, the graph of the equation must lie on or above the horizontal axis for all possible values of the random variable.

normal probability distribution

the relative frequency histogram of the random variable has the shape of a normal curve

inflection points

the points on a curve where the curvature of the graph changes

properties of the normal density curve

1. It is symmetric about its mean
2. Because mean = median = mode, there is a single peak at the mean
3. It has inflection points at 1 standard deviation of the mean
4. The area under the curve is 1
5. The area under the curve to the right of the mean and the area under the curve to the left of the mean both equal .5
6. As x increases and decreases, the graph approaches but never reaches the horizontal axis.
7. The Empirical rule applies

model

an equation, table, or graph that is used to describe reality

standard normal distribution

The normal distribution with mean µ = 0 and standard deviation σ = 1. Its ordinary scores are the same as its z-scores.

P(a < Z < b)

the probability that a standard normal random variable is between a and b

P(Z > a)

the probability that a standard normal random variable is greater than a

P(Z > a)

the probability that a standard normal random variable is less than a

procedure for finding the value of a normal random variable corresponding to a specified proportion, probability, or percentile

1. Draw a normal curve and shade the area corresponding to the proportion, probability, or percentile
2. Look up the z-score that corresponds to the shaded area
3. Obtain the normal value from the formula x = µ + zσ

normal probability plot

a graph that plots observed data versus normal scores

normal score

the expected z-score of the data value, assuming that the distribution of the random variable is normal

drawing a normal probability plot

1. Arrange the data in ascending order
2. Compute fi = (i – 0.375)/(n + 0.25), where i is the index (the position of the data value in the ordered list) and n is the number of observations. The expected proportion of observations less than or equal to the ith data value is fi.
3. Look up the z-score corresponding to fi.
4. Plot the observed values on the horizontal axis and the corresponding expected z-scores on the vertical axis.

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