This module provides a review of the binomial and uniform probability distribution functions and their properties.
Formula
Binomial
~
= the number of successes in
independent trials
= the number of independent trials
takes on the values
0,1, 2, 3, ...,
= the probability of a success for any trial
= the probability of a failure for any trial
The mean is
. The standard deviation is
.
Formula
= a real number between
and
(in some instances,
can take on the
values
and
).
= smallest
;
= largest
~
The mean is
The standard deviation is
Probability density function:
for
Area to the Left of x:
Area to the Right of x:
Area Between c and d:
.
Glossary
Bernoulli trials
An experiment with the following characteristics:
(1). There are only 2 possible outcomes called “success” and “failure” for each trial.
(2). The probability p of a success is the same for any trial (so the probability q = 1–p of a failure is the same for any trial
Binomial distribution
A discrete random variable (RV) which arises from Bernoulli trials. There are a fixed number, n, of independent trials. “Independent” means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV
X is defined as the number of successes in n trials. The notation is:X~ B
(n
,p
). The mean is μ=npand the standard deviation is
=
. The probability of exactly x successes in n trials is
P(X = x) = (
)
Conditional probability
The likelihood that an event will occur given that another event has already occurred.
Expected value
Expected arithmetic average when an experiment is repeated many times. (Also called the mean). Notations:
E(x),μ.For a discrete random variable (RV) with probability distribution functionP(x),the definition can also be written in the formE(x)
=μ=∑xP(x).
Exponential distribution
A continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital. Notation: X~Exp(m). The mean is μ =
and the standard deviation is σ =
. The probability density function is ƒ(x) =
,
and the cumulative distribution function is P(
) = 1 -
.
Geometric distribution
A discrete random variable (RV) which arises from the Bernoulli trials. The trials are repeated until the first success. The geometric variable X is defined as the number of trials until the first success. Notation:
X∼G(p) . The mean is μ =
is The probability of exactly x failures before the first success is given by the formula:
Hypergeometric distribution
A discrete random variable (RV) that is characterized by:
(1). A fixed number of trials.
(2). The probability of success is not the same from trial to trial.
We sample from two groups of items when we are interested in only one group.
X is defined as the number of successes out of the total number of items chosen. Notation:X~H(r,b,n)., where r = the number of items in the group of interest, b = the number of items in the group not of interest, and n = the number of items chosen.
Mean
A number that measures the central tendency. A common name for mean is 'average.' The term 'mean' is a shortened form of 'arithmetic mean.' By definition, the mean for a sample (denoted by
) is
=
and the mean for a population (denoted byμ) is μ =
Poisson distribution
A discrete random variable (RV) that counts the number of times a certain event will occur in a specific interval. Characteristics of the variable:
(1). The probability that the event occurs in a given interval is the same for all intervals.
(2). The events occur with a known mean and independently of the time since the last event.
The distribution is defined by the mean μ of the event in the interval. Notation:
X~P(μ). The mean is μ = np. The standard deviation is
. The probability of having exactly x successes in r trials is P(X = x) =
Probability distribution function (pdf)
A mathematical description of a discrete random variable (RV), given either in the form of an equation (formula) , or in the form of a table listing all the possible outcomes of an experiment and the probability associated with each outcome.
Example:
A biased coin with probability 0.7 for a head (in one toss of the coin) is tossed 5 times. We are interested in the number of heads (the RV X = the number of heads). X is Binomial, so
X∼B(
5,
0.
7) and
P
(X
=x
)= (
).
or in the form of the table:
|
|
|
| 0 |
0.0024 |
| 1 |
0.0284 |
| 2 |
0.1323 |
| 3 |
0.3087 |
| 4 |
0.3602 |
| 5 |
0.1681 |
Random variable (rv)
See
variable .
A continuous random variable (RV) that has equally likely outcomes over the domain,
. Often referred as the
Rectangular distribution because the graph of the pdf has the form of a rectangle. Notation:
X~U(a,b). The mean is μ =
and the standard deviation is
. The probability density function is ƒ(x) =
for
or
. The cumalative distribution is P(
) =
.
Variable (random variable)
A characteristic of interest in a population being studied. Common notation for variables are upper case Latin letters
X,Y,
Z,...; common notation for a specific value from the domain (set of all possible values of a variable) are lower case Latin lettersx,
y,z,.... For example, if
X is the number of children in a family, thenx represents a specific integer 0, 1, 2, 3, .... Variables in statistics differ from variables in intermediate algebra in two following ways.
(1). The domain of the random variable (RV) is not necessarily a numerical set; the domain may be expressed in words; for example, if
X = hair color then the domain is {black, blond, gray, green, orange}.
(2.) We can tell what specific value
x of the Random VariableX takes only after performing the experiment.
