Stats 2B03: Statistical Methods for Science
Chapter 4: Probability Distributions
4.2 Probability Distributions of Discrete Variables
 The probability distribution of a discrete random variable is a table, graph,
formula, or other device used to specify all possible values of a discrete
random variable along with their respective probabilities

 ∑ for all x
 Cumulative distributions: may be obtained by successively adding the
probabilities. Graph of a cumulative probability distribution is called an ogive
 Mean and variance of discrete probability distributions:
∑ ∑ ∑ , where p(x) is the relative
frequency of a given random variable X. the standard deviation is simply the
positive square root of the variance
4.3 The Binomial Distribution
 When a random process or experiment, called a trial, can result in only one of
two mutually exclusive outcomes, the trial is called a Bernoulli trial
 The Bernoulli process: a sequence of Bernoulli trials forms a Bernoulli
process under the following conditions:
Each trial results in one of two possible, mutually exclusive, outcomes.
One of the possible outcomes is denoted (arbitrarily) as a success, and
the other is denoted as a failure
The probability of a success, denoted by p, remains constant from trial
to trial. The probability of a failure, 1 – p, is denoted by q
The trials are independent; that is, the outcome of any particular trial
is not affected by the outcome of any other trial
 Large sample procedure – use of combinations: a combination of n
objects taken x at a time is an unordered subset of c of the n objects,
n
 Binomial table: gives the probability that X is less than or equal to some
specified value. Give the cumulative probabilities from x=0 up through some
specified positive number of successes
 Using table B when p>.5:  
 The binomial parameters: n and p are parameters in the sense that they
are sufficient to specify a binomial distribution
4.4 The Poisson Distribution
 If x is the number of occurrences of some random even in an interval of time
or space, the probability that x will occur is given by
 The poisson process: binomial distribution results from a set of
assumptions about an underlying process yielding a set of numerical
observations: The occurrences of the events are independent. The occurrence of an
event in an interval of space or time has no effect on the probability of
a second occurrence of the event in the same, or any other, interval.
Theoretically, an infinite number of occurrences of the event must be
possible in the interval.
The probability of the single occurrence of the event in a given
interval is proportional to the length of the interval
In any infinitesimally small portion of the interval, the probability of
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