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S1 Key Points – Probability Distributions
Random Variables
A random variable is a quantity whose value depends on chance. For all probability
distributions the sum of all probabilities is 1.
The expectation and variance of a random variable X are given by:
E( X )     xi pi
Var ( X )   2   xi pi   2
2
The Binomial and Geometric Distributions
For both the binomial and geometric distributions a single trial has exactly two
possible outcomes (success and failure) and these are mutually exclusive. The
probability of success, p , and the failure, q  1  p , are constant and independent of
the outcomes of other trials.
For the binomial distribution there are a fixed number of trials, n . When
X ~ B (n, p ) the random variable, X , which represents the number of successes in
the n trials of this experiment has probabilities given by:
 n
P X  x     p x (1  p) n  x  n C x p x q n  x
 x
For cumulative probabilities you can use the table, which show values for P X  x 
The expectation and variance of the binomial distribution are given by:
E ( X )  np
Var ( X )  np(1  p)  npq
For the geometric distribution trials are repeated until a success occurs. When
X ~ Geo( p ) the probability that the random variable, X , which represents the
number of trials up to and including the first success has probabilities given by:
P X  x   p(1 p) x1  pq x 1
P X  x   (1  p) x1  q x 1
P X  x   1 (1 p) x  1 q x
The expectation of the geometric distribution is given by:
E( X ) 
1
p