5 Discrete Random Variables
Definition: random variable
A quantitative variable whose value depends on chance.
Definition: discrete random variable (DRV)
Possible values form a finite or countably infinite set.
Notation: {x = n}
- event that x is n
P(x = n) - probability that x is n
Definition: Probability Distribution
Listing of possible values and
corresponding probabilities of a DRV, or a formula for the probabilities.
Definition: Probability histogram
Possible values of DRV on horizontal axis, probabilities of their occurrence on vertical axis.
Theorem: For any DRV x, 
Definition: Mean of DRV =
also called expected value, or expectation.
Definition: Standard Deviation of DRV = 
Shortcut =
Definition: 
for
Definition: 
Definition: 
is the
binomial coefficient if n is a positive integer and x is a nonnegative integer.
Definition : Repeated identical trials are Bernoulli Trials if:
i. each trial has 2 outcomes, called success (s) and failure (f).
ii. trials are independent
iii. the probability of success (p) remains identical through each trial
(Note: condition iii. requires replacement)
Theorem: Sampling without
replacement from a 2-category population can be considered a Bernoulli trial,
provided the sample size is small relative to the population size (≤ 5%).
Theorem: In n Bernoulli trials, the number of outcomes containing exactly x successes is equal to
.
Theorem: Suppose n Bernoulli trials are to be performed,
with probability of success on any trial p. Let x denote the total number of successes in n trials.
Then: 
where x is called the binomial random variable
with binomial distribution with parameters n and p.
Procedure: To find a Binomial Probability Formula
1. Identify a success
2. Determine p.
3. Determine n.
4. Use P(x) (x = number of outcomes).
Theorem: If x is a binomial
random variable,
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