Discrete Random Variables - Stats Module 3

includes binomial distribution

random variable

A ___ ____ X is a function that associates numbers with some attributes of a sample outcome of an experiment.

• You can have more than one randome variable.

• Either discrete or continuous

relation

relationship between x and y coords. It maps inputs to outputs. e.g. y² = x

function

any input has exactly one output. One output y per input x. e.g. If x=4, then y can only be 7.

• One to one

• Many to One

probability distribution

_____ _____ for a discrete random variable X is a formula, table, or graph that gives all possible values k of X, and the probability px(k) = P(X = k) associated with each value

• These values k are mutually exclusive events

• For a discrete probability distribution, see the attached properties

probability distribution table

cumulative distribution

_____ _____ for a discrete random variable X is a formula, table, or graph that gives all the possible values k and Fx(k) = P(X </= k), the probability that X is at most k.

cumulative distribution table

mean or expected value

Let X be a discrete random variable with probability distribution px(k). The

___ of X, which is denoted by (see attached) or E(X), is given as

standard deviation

Let X be a discrete random variable with probability distribution px(k). The ___ of X, which is denoted by (see attached), is given as

variance

Let X be a discrete random variable with probability distribution px(k). Attached is the square of the standard deviation (just take out the sqr root to solve for it)

binomial experiment

conditions:

1. the experiment consists of n identical trials (e.g. imagine three distinguishable coins being tossed)

2. results in one of two outcomes: success or failure.

3. probability of success on a single trial: p

   probability of failrue: q = 1-p

4. The trials are independent.

bernoulli trial

trials in binomial experiments

binomial random variable X

the number of successes in a binomial experiment. For the special case when n=1, we also call X a Bernoulli random variable.

probability of k successes

Let X be the binomial random variable corresponding to a binomial experiment consisting of n independent Bernoulli trials. If p is the probability of success in each trial, then the ____ is

for k = 0, 1, …, n

theorem 2

for the mean, variance, and standard deviation of X

where n = number of trials and p = probability of success