Discrete distribution  (Page 2/2)

A better understanding of a particular probability distribution can often be obtained with a graph that depicts the p.d.f. of X .

Two types of graphs can be used to give a better visual appreciation of the p.d.f., namely, a bar graph and a probability histogram . A bar graph of the p.d.f. f(x) of the random variable X is a graph having a vertical line segment drawn from $\left(x\mathrm{,0}\right)$ to $\left[x,f\left(x\right)\right]$ at each x in R , the space of X . If X can only assume integer values, a probability histogram of the p.d.f. f(x) is a graphical representation that has a rectangle of height f(x) and a base of length 1, centered at x , for each $x\in R$ , the space of X .

CUMULATIVE DISTRIBUTION FUNCTION

Let define the function F(x) by

$$F\left(x\right)=P\left(X\le x\right)={\displaystyle \sum _{t\in A}f\left(t\right)}.$$

The function F(x) is called the distribution function (sometimes cumulative distribution function ) of the discrete-type random variable X .

Several properties of a distribution function F(x) can be listed as a consequence of the fact that probability must be a value between 0 and 1, inclusive:

• $0\le F(x)\le 1$ because F(x) is a probability,
• F(x) is a nondecreasing function of x ,
• $F\left(y\right)=1$ , where y is any value greater than or equal to the largest value in R ; and $F\left(z\right)=0$ , where z is any value less than the smallest value in R ;
• If X is a random variable of the discrete type, then F(x) is a step function, and the height at a step at x , $x\in R$ , equals the probability $P\left(X=x\right)$ .

MATHEMATICAL EXPECTATION

If f(x) is the p.d.f. of the random variable X of the discrete type with space R and if the summation

${\displaystyle \sum _{R}u\left(x\right)f\left(x\right)={\displaystyle \sum _{x\in R}u\left(x\right)f\left(x\right)}}$

exists, then the sum is called the mathematical expectation or the expected value of the function u(X) , and it is denoted by $E\left[u\left(X\right)\right]$ . That is,

$$E\left[u\left(X\right)\right]={\displaystyle \sum _{R}u\left(x\right)f\left(x\right)}.$$

We can think of the expected value $E\left[u\left(X\right)\right]$ as a weighted mean of u(x) , $x\in R$ , where the weights are the probabilities $f\left(x\right)=P\left(X=x\right)$ .

There is another important observation that must be made about consistency of this definition. Certainly, this function u(X) of the random variable X is itself a random variable, say Y . Suppose that we find the p.d.f. of Y to be g(y) on the support $R_1$ . Then E(Y) is given by the summation ${\displaystyle \sum _{y\in R_1}yg\left(y\right)}$

In general it is true that $${\displaystyle \sum _{R}u\left(x\right)f\left(x\right)}={\displaystyle \sum _{y\in R_1}yg\left(y\right)};$$ that is, the same expectation is obtained by either method.

When it exists, mathematical expectation E satisfies the following properties:

• If c is a constant, E ( c )= c ,
• If c is a constant and u is a function, $E\left[cu\left(X\right)\right]=cE\left[u\left(X\right)\right]$ ,
• If $c_1$ and $c_2$ are constants and $u_1$ and $u_2$ are functions, then $E\left[c_1{u}_1\left(X\right)+c_2{u}_2\left(X\right)\right]=c_{1}E\left[u_1\left(X\right)\right]+c_{2}E\left[u_2\left(X\right)\right]$

First, we have for the proof of (1) that

$E\left(c\right)={\displaystyle \sum _{R}cf\left(x\right)=c{\displaystyle \sum _{R}f\left(x\right)}}=c$

because ${\displaystyle \sum _{R}f\left(x\right)=1.}$

Next, to prove (2), we see that

$E\left[cu\left(X\right)\right]={\displaystyle \sum _{R}cu\left(x\right)f\left(x\right)=c{\displaystyle \sum _{R}u\left(x\right)f\left(x\right)=cE\left[u\left(X\right)\right]}}.$

Finally, the proof of (3) is given by

$E\left[c_1{u}_1\left(X\right)+c_2{u}_2\left(X\right)\right]={\displaystyle \sum _R\left[c_1{u}_1\left(x\right)+c_2{u}_2\left(x\right)\right]}f\left(x\right)={\displaystyle \sum _R{c}_1{u}_1\left(x\right)f\left(x\right)+{\displaystyle \sum _R{c}_2{u}_2\left(x\right)f\left(x\right)}}.$

By applying (2), we obtain

$E\left[c_1{u}_1\left(X\right)+c_2{u}_2\left(X\right)\right]=c_{1}E\left[u_1\left(x\right)\right]+c_{2}E\left[u_2\left(x\right)\right].$

Property (3) can be extended to more than two terms by mathematical induction; That is, we have

3'. $E\left[{\displaystyle \sum _{i=1}^k{c}_i{u}_i\left(X\right)}\right]={\displaystyle \sum _{i=1}^k{c}_{i}E\left[u_i\left(X\right)\right]}.$

Because of property (3’), mathematical expectation E is called a linear or distributive operator .