Statistics Chapter 4

Probability Distribution (of a DRV)

a graph, table, or formula that specifies the probability associated with each possible value the random variable can assume

Random Variable

a variable that assumes numerical values associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample point

Probability Density Function for a Uniform Random Variable

f(x) = 1 / d - c c less than or equal too "x" less than or equal too d

Mean for a Uniform Random Variable

mu = ( c + d ) / 2

Probability Distribution for a Poisson Random Variable

mu is actually lambda here

Binomial Probability Distribution

p= Probability of a success on a single trial q = 1 - p n = Number of Trials x = Number of successes in n trials n - x = Number of failures in n trials

Standard Deviation (of a DRV)

square root of variance (σ^2)

Binomial Experiment

when an experiment results in dichotomous responses. example: tossing a coin. It has binomial variables

Graphical Form of a Discrete Variable

x-axis are the possibilities y-axis are the probabilities

Graphical Form of a Continuous Variable

x-axis are the values of the possibilities y-axis are the probability densities

Mean/ Expected Value (of a DRV)

μ (mu) = E(x) = Sum of (x*p(x))

Variance (of a DRV)

σ^2 = = E [(x- μ )^2] = Sum of (x - μ )^2p(x)

Uniform Frequency Function

(Base)(Height)= (d - c) (1 / d - c) = 1

Characteristics of a Poisson Random Variable

1. The experiment consists of counting the number of times a certain event occurs during a given unit of time or in a given area or volume (weight, distance, or any other unit of measurement) 2. The probability that an event occurs in a given unit of time, area, or volume is the same for all the units. 3. The number of events that occur in one unit of time, area, or volume is independent of the number that occur in any other mutually exclusive unit. 4. The mean (or expected) number of events in each unit is denoted by the Greek letter lambda, λ.

Characteristics of a Binomial Experiment

1. The experiment consists of n identical trials 2. There are only two possible outcomes on each trial. We will denote one outcome by S (for success) and the other by F (for failure) 3. The probability of S remains the same from trial to trial. This probability is denoted by p, and the probability of F is denoted by q. Note that q = 1 - p. 4. The trials are independent 5. The binomial random variable x is the number of S's in n trials.

Poisson Distribution

A type of discrete variable probability distribution that is often useful in describing the number of rare events that will occur in a specific period of time or in a specific area or volume.

Probability Distribution for a Uniform Random Variable

Continuous random variables that appear to have equally likely outcomes over their range of possible values.

What distributions will be on the test:

Discrete: poisson, binomial Continuous: uniform, or normal Bring Binomial and Normal Tables

Continuous

Random Variables that can assume values are corresponding to any of the points contained in one or more intervals (i.e, values that are infinite and uncountable)

Discrete

Random variables that can assume a countable number (finite or infinite) of values are called discrete.

standard deviation for a Uniform Random Variable

St.D = (d - c)\ √12

What is a standard normal Curve?

Standard Normal Distribution is a normal distribution with μ = 0 and

Requirements for the Probability Distribution of a DRV

1. p(x) greater than or equal too 0 for all values of x 2. Sum of p(x) = 1