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Roll a four-sided die twice and let X equal the larger of the two outcomes if there are different and the common value if they are the same. The sample space for this experiment is S = [ ( d 1 , d 2 ) : d 1 = 1,2,3,4 ; d 2 = 1,2,3,4 ] , where each of this 16 points has probability 1/16. Then P ( X = 1 ) = P [ ( 1,1 ) ] = 1 / 16 , P ( X = 2 ) = P [ ( 1,2 ) , ( 2,1 ) , ( 2,2 ) ] = 3 / 16 , and similarly P ( X = 3 ) = 5 / 16 and P ( X = 4 ) = 7 / 16 . That is, the p. d.f. of X can be written simply as f ( x ) = P ( X = x ) = 2 x 1 16 , x = 1,2,3,4.

We could add that f ( x ) = 0 elsewhere; but if we do not, one should take f(x) to equal zero when x R .

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

the graph of the p.d.f. when f ( x ) > 0 , would be simply the set of points { [ x , f ( x ) ] : x R }, where R is the space 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 ( x ,0 ) to [ x , f ( x ) ] 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 R , the space of X .

CUMULATIVE DISTRIBUTION FUNCTION
Let define the function F(x) by
F ( x ) = P ( X x ) = t A f ( t ) .
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 F ( x ) 1 because F(x) is a probability,
  • F(x) is a nondecreasing function of x ,
  • F ( y ) = 1 , where y is any value greater than or equal to the largest value in R ; and F ( z ) = 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 R , equals the probability P ( X = x ) .
It is clear that the probability distribution associated with the random variable X can be described by either the distribution function F(x) or by the probability density function f(x) . The function used is a matter of convenience; in most instances, f(x) is easier to use than F(x) .

Graphical representation of the relationship between p.d.f. and c.d.f.

Area under p.d.f. curve to a equal to a value of c.d.f. curve at a point a .
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
R u ( x ) f ( x ) = x R u ( x ) f ( x )
exists, then the sum is called the mathematical expectation or the expected value of the function u(X) , and it is denoted by E [ u ( X ) ] . That is,
E [ u ( X ) ] = R u ( x ) f ( x ) .
We can think of the expected value E [ u ( X ) ] as a weighted mean of u(x) , x R , where the weights are the probabilities f ( x ) = P ( X = x ) .
The usual definition of the mathematical expectation of u(X) requires that the sum converges absolutely; that is, x R | u ( x ) | f ( x ) exists.

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 y R 1 y g ( y )

In general it is true that R u ( x ) f ( x ) = y R 1 y g ( y ) ; that is, the same expectation is obtained by either method.

Let X be the random variable defined by the outcome of the cast of the die. Thus the p.d.f. of X is

f ( x ) = 1 6 , x = 1,2,3,4,5,6 .

In terms of the observed value x , the function is as follows

u ( x ) = { 1, x = 1,2,3, 5, x = 4,5, 35, x = 6.

The mathematical expectation is equal to

x = 1 6 u ( x ) f ( x ) = 1 ( 1 6 ) + 1 ( 1 6 ) + 1 ( 1 6 ) + 5 ( 1 6 ) + 5 ( 1 6 ) + 35 ( 1 6 ) = 1 ( 3 6 ) + 5 ( 2 6 ) + 35 ( 1 6 ) = 8.
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Let the random variable X have the p.d.f. f ( x ) = 1 3 , x R , where R ={-1,0,1}. Let u ( X ) = X 2 . Then

x R x 2 f ( x ) = ( 1 ) 2 ( 1 3 ) + ( 0 ) 2 ( 1 3 ) + ( 1 ) 2 ( 1 3 ) = 2 3 .

However, the support of random variable Y = X 2 is R 1 = ( 0,1 ) and

P ( Y = 0 ) = P ( X = 0 ) = 1 3 P ( Y = 1 ) = P ( X = 1 ) + P ( X = 1 ) = 1 3 + 1 3 = 2 3 .

That is, g ( y ) = { 1 3 , y = 0, 2 3 , y = 1 ; and R 1 = ( 0,1 ) . Hence

y R 1 y g ( y ) = 0 ( 1 3 ) + 1 ( 2 3 ) = 2 3 , , which illustrates the preceding observation.

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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 [ c u ( X ) ] = c E [ u ( X ) ] ,
  • If c 1 and c 2 are constants and u 1 and u 2 are functions, then E [ c 1 u 1 ( X ) + c 2 u 2 ( X ) ] = c 1 E [ u 1 ( X ) ] + c 2 E [ u 2 ( X ) ]

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

E ( c ) = R c f ( x ) = c R f ( x ) = c

because R f ( x ) = 1.

Next, to prove (2), we see that

E [ c u ( X ) ] = R c u ( x ) f ( x ) = c R u ( x ) f ( x ) = c E [ u ( X ) ] .

Finally, the proof of (3) is given by

E [ c 1 u 1 ( X ) + c 2 u 2 ( X ) ] = R [ c 1 u 1 ( x ) + c 2 u 2 ( x ) ] f ( x ) = R c 1 u 1 ( x ) f ( x ) + R c 2 u 2 ( x ) f ( x ) .

By applying (2), we obtain

E [ c 1 u 1 ( X ) + c 2 u 2 ( X ) ] = c 1 E [ u 1 ( x ) ] + c 2 E [ u 2 ( x ) ] .

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

3'. E [ i = 1 k c i u i ( X ) ] = i = 1 k c i E [ u i ( X ) ] .

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

Let X have the p.d.f. f ( x ) = x 10 , x =1,2,3,4.

then

E ( X ) = x = 1 4 x ( x 10 ) = 1 ( 1 10 ) + 2 ( 2 10 ) + 3 ( 3 10 ) + 4 ( 4 10 ) = 3 E ( X 2 ) = x = 1 4 x 2 ( x 10 ) = 1 2 ( 1 10 ) + 2 2 ( 2 10 ) + 3 2 ( 3 10 ) + 4 2 ( 4 10 ) = 10,

and

E [ X ( 5 X ) ] = 5 E ( X ) E ( X 2 ) = ( 5 ) ( 3 ) 10 = 5.

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Source:  OpenStax, Introduction to statistics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10343/1.3
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