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Chapter 4. Discrete Probability Distributions

Section 4.1. Random Variables and Their Probability Distributions

Jiaping Wang

Department of Mathematical Science

02/04/2013, Monday

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Outline

Random Variables and Probability Functions

Distribution Functions Examples

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Part 1. Random Variables and Probability Functions

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Random Variable

For example, flip a fair coin, denote X=0 meaning tail, X=1 meaning head. So P(X=0) = P(X=1) = ½ and P(X=n)=0 for n≠ 0 or 1 . Then the probability P(X≤2)=P(X=0)+P(X=1)+P(X=2)=1.

A random variable or stochastic variable is a variable  whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense). As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability.

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Definition 4.1 A random variable is a real-valued function whose domain is a sample space.

Definition 4.1

The basic concept of "random variable" in statistics is real-valued. However, one can consider arbitrary types such as boolean values, categorical variables, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, functions, and processes. 

The random variable can be either continuous or discrete.

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Definition 4.2

A random variable X is said to be discrete if it can take on only a finite number – or a countably infinite number – of possible values x. The probability function of X, denoted by p(x), assigns probability to each value x of X so that the following conditions hold:

1. P(X=x)=p(x)≥0;2. ∑ P(X=x) =1, where the sum is over all possible values of x.

The probability function is sometimes called the probability mass function of X to denote the idea that a mass of probability is associated with values for discrete points.

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Cont.

It is often convenient to list the probabilities for a discrete random variable in a table. See following example.

x p(x)0 0.041 0.322 0.64Total 1.00

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Example 4.1

A local video store periodically puts its used movie in a bin and offers to sell them to customers at a reduced price. Twelve copies of a popular movie have just been added to the bin, but three of these are defective. A customer randomly selects two of the copies for gifts. Let X be the number of defective movies the customer purchased. Find the probability function of X and graph the function.

Denote the D as the defective, ND as the non-defective. There are two step selections.

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Cont.

x p(x)0 72/1321 54/1322 6/132Total 1

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Part 3. Distribution Function

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Definition 4.3

The distribution function F(b) for a random variable X isF(b)=P(X ≤ b);

If X is discrete,

Where p(x) is the probability function. The distribution function is often called the cumulative

distribution function (CDF).

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Cont.

P(X≤0)=0.04, P(X<0)=0, P(X ≤1)=P(X=0)+P(X=1)=0.36 but P(X<1)=0.04P(X ≤2)=P(X=0)+P(X=1)+P(X=2)=1.0P(X<2)=P(X=0)+P(X=1)=0.36P(X>2)=1.0P(X ≤ 1.5)=P(X ≤ 1.9)=P(X ≤ 1)=0.36.

x p(x)0 0.041 0.322 0.64Total 1.00

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Cont.

Note from last example, we can find F(x) is a right-continuous function but not left-continuous, that is

Any function satisfies the following 4 properties is a distribution function:

1.

2.

3. The distribution function is a non-decreasing function: if a<b, then F(a)≤ F(b). The distribution function can remain constant, but it can’t decrease as we increase from a to b.

4. The distribution function is right-hand continuous:

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Example 4.2

A large university uses some of the student fees to offer free use of its health center to all students. Let X be the number of times that a randomly selected student visits the center during a semester. Based on historical data, the distribution function of X is given as

1. Graph F.2. Verify that F is a distribution function.3. Find the probability function associated with F.

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Cont.

1. Because F is zero for all values less than zero, so 2. As F is one for all values larger than 3, 3. As x increases, F(x) either remains constant or increases, it means F(x) is

nondecreasing. 4. There are three jumping points: 0, 1, 2 and 3, we can show for each point, F

is right-hand continuous. For example, when h 0+, F(2+h)0.95=F(2).

x p(x)0 0.6-0=0.61 0.8-0.6=0.22 0.95-0.8=0.153 1.0-.095=0.05