Discrete Random Variables and Probability Distributions

probability distribution function

Assigns a probability to each value of X f(Xi) = P(X = xi)

How many random variables can be defined in a sample space?

More than one.

Find the probability that the first and second camera pass the test, and the third one fails. - The probability that a camera passes the test is 0.8

P(ppf) = (0.8)(0.8)(0.2) = 0.128

How to verify that a function is a probability mass function (pmf)?

The sum of P(x) = 1

For the following exercises, determine the range (possible values) of the random variable. An electronic scale that displays weights to the nearest pound is used to weigh packages. The display shows only five digits. Any weight greater than the display can indicate is shown as 99999. The random variable is the displayed weight.

X = display weight on electronic scale Range = {0, 1, 2, ... , 99999}

probability distribution

a description of how the probabilities are distributed over the values of the random variable

discrete random variables

a fixed set of possible x values (whole numbers)

Probability Mass Function (PMF)

a function that gives the probability that a discrete random variable is exactly equal to some value P(X=x)

variance

a measure of the dispersion, or variability in the distribution

Cumulative Distribution Function (CDF)

distribution function, defines the probability that a random variable, X, takes on a value equal to or less than a specific value, x. Represents the sum, or cumulative value, of the probabilities of the outcomes up to and including a specific outcome. F(x) = P(X ≤ x)

mean

measure of the center or middle of the probability distribution

how is the probability of an event generally determined?

p = # of favorable events/ total number of events

Discrete Uniform Distribution

the distribution has a finite number of specified values, each value is equally likely, the distribution is symmetric f(Xi) = 1/n