23. Discrete Probability Distributions

A discrete probability distribution shall be understood as a probability distribution characterized by a probability mass function. Thus, the distribution of a random variable X is discrete, and X is then called a discrete random variable, if

as u runs through the set of all possible values of X. It follows that such a random variable can assume only a finite or countably infinite number of values.

Def: Let X be a random variable that takes the numerical values X1, X2, ..., Xn with probablities p(X1), p(X2), ..., p(Xn) respectively. A discrete probability distribution consists of the values of the random variable X and their corresponding probabilities P(X).  The probabilities P(X) are such that 

∑ P(X) = 1

Cumulative distribution

Def : For every real number x, the cumulative distribution function of a real-valued random variable X is given by

where the right-hand side represents the probability that the random variable X takes on a value less than or equal to x. The probability that X lies in the interval (a, b], where a  <  b, is therefore

Here the notation (a, b], indicates a semi-closed interval.

If treating several random variables X, Y, ... etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is omitted. It is conventional to use a capital F for a cumulative distribution function, in contrast to the lower-case f used for probability density functions and probability mass functions. This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the normal distribution.

The CDF of a continuous random variable X can be defined in terms of its probability density function ƒ as follows:

F(x) has the following properties:

1. F(x) non-decreasing: F(x1)≤F(x2), for x1≤x2

2.

3. F(x) is continuous from the right Ɏh=R (x→+0) lim F(x+h)=F(x)

24. Continuous distribution function

Continuous distribution function-random variables that can take on values on a continuous scale

Def 1: the function f(x) is a probability density function for the continuous random variable x define over the set of real number

1. f(x) ≥0

2. 

3. P(a<x<b)=

Def 2: the cumulative distribution function F(x) of continuous random variable x with density function f(x)

1. F(x)=p(X≤x)= where -∞<x<∞

Properties:

1. a<x<b P(a<x<b)=F(b)-F(b)

2. f(x) =

mean

µ=ᶋ x f(x)dx

standard deviation

where f(x)- is density function

σ² = E((X-µ₀²))=ᶋ((x-µ) ²f(x)dx) where ᶋ -∞<x<∞

Probability density function 

In probability theory, a probability density function, or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value. The probability for the random variable to fall within a particular region is given by the integral of this variable’s density over the region. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.

A random variable X with values in a measure space   (usually Rⁿ with the Borel sets as measurable subsets) has as probability distribution the measure X_∗P on  : the density of X with respect to a reference measure μ on   is the Radon–Nikodym derivative:

That is, f is any measurable function with the property that:

for any measurable set  .