Section 1 : Discrete Random Variables(1)
 

[probability function][cumulative distribution fn.]

 

 

 

Concept

The basic idea is assign a real number xr to every event Er happening in the sample space S .

Example

Consider the case when a coin is tossed twice. The possible results( the Sample Space S) are:

(T T)   (T H)   (H T)   (H H)

Assigning numbers for the number of 'heads' occuring:

(0)   (1)   (1)   (2)

So overall, there are 3 possible outcomes:

no heads     one head     two heads

This can be written as the result:

0               1              2

So the probability of flipping a coin twice and obtaining a head is:

no. of heads
sample space S
probability
0
(T T)   (T H)   (H T)   (H H)
1/4
1
(T T)   (T H)   (H T)   (H H)
1/2
2
(T T)   (T H)   (H T)   (H H)
1/4

The Probability (Density) Function PDF

The probability function of X, written as P(X=x), is used to allocate probabilities.

We describe X as a discrete random variable, when it has a finite number of possible values and if the sum of probabilities is one.

X takes the value xr when the event Er occurs.

From our example(above) it is self evident that:

   event E = 1     one head    x = 1      P(X=1) = 1/2

So all the results in our example could be displayed as:

event E
x
P(X=x)
0
0
(T T)   (T H)   (H T)   (H H)
1/4
1
1
(T T)   (T H)   (H T)   (H H)
1/2
2
2
(T T)   (T H)   (H T)   (H H)
1/4

The probability function can also be written as p(x) .

therefore                    P(X=x) = p(x)

 

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Cumulative Distribution Function CDF

F(x) cumulative distribution function CDF

This function returns the sum of probabilities of X for values of x less than or equal to x0 .

It also may be described as the function F(x0) giving the probability of X when its value is less than or equal to x0 .

A more general definition is that cumulative probability F(x0) is the value a random variable takes when it falls between a specific range.

Example #1

Consider a situation where there are four possible outcomes(x = 0, 1, 2, 3 ), with different probabilities for each of these.
The probability that X is less than or equal to 3 is given by:

cumulative distribution function CDF example #1

Example #2 (with reference to the coin flip table above)

A coin is flipped twice. What is the probability that 1 head or no heads result?

cumulative probability distributions - example #2

All the results for heads(x) from 2 coin flips can be summarized in a modified table:

E
x
P(X=x)
P(X<x)
0
0
(T T)   (T H)   (H T)   (H H)
1/4
1/4
1
1
(T T)   (T H)   (H T)   (H H)
1/2
3/4
2
2
(T T)   (T H)   (H T)   (H H)
1/4
1

Notice how P(X<x) is derived from P(X=x) when x is incremented.

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