Math 3200-- Exam 2 Content

random variable

a function that assigns a probability to each specific outcome in the sample space of an experiment

discrete random variable

a variable that can only take on integer values. Its sample space is finite or countably infinite (like it won’t take that long to count, or it is countable)

Examples: number of children in a household

Lower bound is included in discrete variables

continuous random variable

a variable that can take on any value in any interval-- values do not have to be integers.

Examples: height, weight, pH

probability distribution

shows you how the probabilities are distributed for all the values in the sample space

Example: normal distribution, CDF, PDF

probability mass function

a probability distribution for discrete random variables. Would be used to express the probabilities of the outcomes of tossing a dice because you can only either roll a 1,2,3,4,5, or 6-- discrete variables

cumulative distribution function (CDF)

an example of a probability distribution; tells you the probability of that variable being less then or equal to x. Denoted as F(x)

Can be used for continuous or discrete variables

Only increases because it is cumulative-- adds up everything before it

probability distribution function (PDF)

a PMF but for continuous random variables only. The probability is the area under the curve. Denoted as f(x)

To get the CDF from PDF: integrate the PDF

To get the PDF from the CDF: take the derivative of the CDF

expected values

Discrete: multiply each value by its probability and add them up (would get from PMF)

Continuous: integrate x multiplied by f(x) (get from PDF)

Function of a Random variable: use that integral of h(x) and f(x)

formula for variance

only one for all distributions

population percentiles

F(xa) = P(X<xa) = 1 -a

Bernoulli Distribution

used for discrete random variables; in a Bernoulli trial, the outcome is either a success or a failure

If the outcome is a success, value is 1

If the outcome is a failure, value is 0

binomial distribution

used for discrete random variables; many Bernoulli trials, but you count the number of successes in each trial

Each trial performed independently

hypergeometric distribution

used for discrete random variables; each thing in the population is denoted as one thing-- either M1 or M2. The hypergeometric variable measures how many things are labeled M1 in the sample.

Sampling occurs **without replacement**, so events are no longer independent (because the sample size decreases as you sample without replacement)

geometric distributions

used for discrete random variables; represents the amount of trials it takes until you get the first success. Is a type of negative binomial distribution

Independent trials

negative binomial distribution

used for discrete random variables; represents the amount of trials it takes until you get an rth success

Independent trials

poisson distribution

used for discrete random variables; represents the probability of a number of events occurring is a specific interval

exponential distribution

used for continuous random variables; represents lifetimes of equipment or waiting times until an event occurs

Has a CDF and a PDF

normal distribution

used for continuous random variables

Z is a normal random variable with mew=0 and variance=1

binomial distributions converging to normal distributions

with very large sample sizes, binomial and exponential distributions converge to normal distributions.

If n is larger than 30, you can approximate the data using a normal distribution

But, median, min, and max do not converge to normal. Only the mean and variance go to normal distributions.

z score

x-muw/standard deviation

what makes something a valid joint pdf

the product of the integrals of the marginal distributions equals 1

what makes something a valid cdf

non-decreasing, and the lower bound is 0 and upper bound is 1

what makes two marginal distributions independent

they have two different ranges, and you can separate them out-- you can factor them out into 2 separate distributions, without the terms affecting each other