Discrete Probability Distributions

• A sample space can be difficult to describe and work with if its elements are not numeric.

• Random Variable• A random variable is a function that assigns

each element in the sample space to a number.• The random variable X has range:

• {x|x=X(s), for all s in S}.• More than one random variable can be

associated with an experiment.

Discrete Random Variable A discrete random variable is a random

variable that has a finite or countable sample space. A sample space that can be mapped to the

integers is said to be countably infinite (or countable).

Probability Distribution The probability distribution of a rv describes

the distribution of total probability to all possible values of the rv.

A discrete probability distribution, called a probability mass function (pmf), specifies the probability of each distinct element in the sample space.

p(x) = P( X = x ) = P(all s in S: X(s)=x)

Properties of p(x)

1. p(x)≥0 for all x in S

2. ∑S p(x) = 1

3. If A is a subset of S, then

P(A) = ∑A p(x)

Note: A pmf can be displayed nicely with a line graph or a probability histogram.

Cumulative Distribution Function (cdf)

The cdf of a discrete random variable X with pmf p(x) is defined for each x as:

F(x) = P( X ≤ x ) = ∑y:y≤x p(y)

For any number x, F(x) is the probability that the rv X will be at most x.

The graph of F(x) for a discrete rv is a step function. For any two numbers a and b with a ≤ b,

P( a ≤ x ≤ b ) = F(b) – F(a-)where a- represents the largest value of X less than a

Mathematical Expectation The mathematical expectation (expected value) of

a discrete rv is the weighted average of all possible values of the rv, where the weight associated with each outcome is its probability.

The Expected Value of X Let X be a discrete rv with pmf p(x). The

expected value (or mean value) of X is:

E[X] = µx = µ = ∑S x · p(x)

Mathematical Expectation The Expected Value of a Function of X

Let X be a discrete random variable with pmf p(x). The expected value of a function h(x) is

E[h(X)] = µh(x) = ∑S h(x) · p(x)

Note that E[h(x)] only exists if ∑S h(x) · p(x) converges, therefore exists.

Properties of E[X] If c is a constant, then E[c] = c.

If c is a constant, then E[cX] = c·E[X].

If c and d are constants, then

E[cX+d] = c·E[X]+d.

If c is a constant and u(x) is a function, then

E[c·u(X)] = c·E[u(X)].

Properties of E[X] Property of a Linear Operator

If ci are constants and ui(x) are functions, then

E[c1·u1(X)+c2·u2(X)+…+cn·un(X)]

= c1·E[u1(X)] + c2·E[u2(X)] + … + cn·E[un(X)]

= ∑i=1,n ci · E[ui(X)]

Variance of Discrete RV

Let X be a discrete random variable with pmf p(x). The Variance of X is

The variance of X measures the amount of spread in the distribution of X.

)()(

])[()(Var2

22

xpx

XEX

S

x

Variance of a Discrete RV

Easier forms for the variance of X include:

22

22

][][

)()(Var

XEXE

xpxXS

Standard Deviation

Let X be a discrete random variable with pmf p(x). The Standard Deviation of X is the square root of the Variance of X.

The standard deviation is commonly used as the measure of spread in a distribution

2)(StDev xxX

Variance of a Function

Let X be a discrete rv with pmf p(x). The variance of a function h(X) is:

2)(

2

2

2)(

2)(

)()(

)()]([)(

)()(Var

XhS

S

Xhxh

xpxh

xpXhExh

XhEXh

Variance of a Function

If a and b are constants and Var(X)=σ2, then

Var( a·X ) = a2 · σ2

Var( X+b ) = σ2

Var( a·X+b ) = a2 · σ2