Random Variables

When a sample space has to be processed in a certain way to create a more appropriate set of data, this process is called a random variable. A random variable maps the sample space onto another set. This other set can be the real number line or an axis in the coordinate plane. This mapping processes also called a function. Random variables are usually described with capital letters such as X, Y or Z. It can be defined as:

Random variable

a function whose domain is the sample space.

An example would be a coin flipped three times. The outcomes would be the sample space S.

S = {	TTT,	TTH,	THT,	HTT,	HTH,	HHT,	THH,	HHH	}
	s1	s2	s3	s4	s5	s6	s7	s8

Counting the number of heads that occur for each element would be a function of the sample space and therefore a random variable.

0 heads	1 head	2 heads	3 heads
X(s1) = 0	X(s2) = 1 X(s3) = 1 X(s4) = 1	X(s5) = 2 X(s6) = 2 X(s7) = 2	X(s8) = 3

As a result, the range of values that the random variable can have is:

X = 0, 1, 2, 3

Probability Density Functions

Discrete

If the range of the random variable contains values that can be counted, it is a discrete random variable.

When probabilities are associated with random variables, the result is a probability density function which is sometimes written as pdf.

For example, the random variable, X, that counts the number of heads in each outcome has the following range of values: 0, 1, 2, 3. Since there are eight outcomes, and only one outcome has 0 heads, the probability density function would produce 1/8 as the probability of getting an outcome was no heads. This would be written as:

P( X = 0 ) = 1/8

The probability for the other outcomes would be:

P( X = 1 ) = 3/8
P( X = 2 ) = 3/8
P( X = 3 ) = 1/8

When placed in the coordinate plane this would look like:

Notice all the probabilities add to 1.

Continuous

If the range of a random variable contains values that are intervals with an infinite number of values, then it is a continuous random variable. Continuous random variables also have pdf's. Note that the value of the pdf itself isn't equal to probability. It is the interval of the pdf that produces a probability value.

If y is a continuous random variable and if the pdf is a function f_Y(y) where y is a value of the random variable. Then the probability of the interval on the pdf is:

P( a ≤ Y ≤ b ) = &int_a^bf_Y(y)dy

Where a and b are endpoints on the interval.

Joint Probability Density Functions

If there are two random variables in the sample space, the pdf that describes both random variables is a joint pdf and for discrete random variables would be noted as:

P_X,Y(x,y) or P(X=x, Y=y)

Where X and Y are the random variables.

For continuous random variables, a joint pdf would correspond to a region on the surface of a plane:

P( a ≤ X ≤ b, c ≤ Y ≤ d) = &int_a^b&int_c^df_X,Y(x,y)dydx

Independent Random Variables

If the random variables are related to independent events, they can be defined as independent random variables as follows:

X and Y are independent if and only if:

f_X,Y(x,y) = f_X(x) * f_Y(y)

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