Chapter 6 (discrete random variables

random variable

is a variable whose value is numerical and is determined by the outcome of an experiment

discrete random variable

when the possible values of a random variable can be counted

or listed (like a dice or a coin)

continuous random variable

When a random variable may assume any numerical value in one or more intervals on the real number line (like any point between 1 and 7 including non integers

a discrete prob-ability distribution

a table, graph, or formula that gives the probability associated

with each possible value that the random variable can assume

We denote the probability distribution of the discrete random variable x as p(x)

p(x) ≥ 0 for each value of x

Σ p(x) = 1

Expected Value or The Mean, of a Discrete Random Variable

The mean, or expected value, of a discrete random variable x is denoted a μx

μx = Σ xp(x)

E.G. 0p(0) + 1p(1) + 2p(2) + 3p(3) + 4p(4) + 5p(5)

variance of a discrete random variable

is equal to the variance formula with the inclusion the probability of each variable

σ²x=∑( (x-μ)² f(x) )

uniform distribution

suppose that a random variable x is

equally likely to assume any one of n

binomial distribution

1. The experiment consists of n identical trials.

2 Each trial results in a success or a failure.

3 The probability of a success on any trial is p and remains constant from trial to trial.

4 The trials are independent (that is, the results of the trials have nothing to do with each other).

Binomial distribution formula

f(x) = [ n! / (x! ( n - x)! ] ( p^x ) (q^ (n-x) )

p = the probability of success

q = the probability of failure = 1 - p

x = the number of succesfuls

n = the amount of trials

f(x) = the probability of x successes in n trials

number of ways to arrange x successes among n trials (exact same as the combination formula)

( n! / (x! ( n - x)! )

x = the number of successes

n = the number of trials

probability of achieving a certain amount of successes and fails in any order

( p^x ) (q^ (n-x) )

The mean of binomial distribution

μx = np

The Variance of standard deviation

σx^2 = npq

Cumulative probability

meaning to add up probabilities of different levels of success to determine probability above or below a certain amount of successes