Random Variables Probability Density Function Cumulative Distribution Function Expectation Variance Sum of Two Random Variables Example

andom Variables

1. Let S be the sample space of an experiment.  A function X that assigns every s Î S a real number x is called a random variable (rv).
2. A random variable X is a discrete random variable (drv) if the sample space S is countable, ie, S is finite or countably infinite.

Thus, X is a discrete random variable if

Notes:

A random variable is denoted by a capital letter, eg X, and the values that it can take by a small letter, eg x.

For example, if X is the rv 'the score obtained when one die is tossed', then X can takes on the values 1, 2, 3, 4, 5, 6, and therefore x = 1, 2, 3, 4, 5 or 6.

'X = 2' is read as 'the score obtained is 2', whereas 'x = 2' means 'x is 2'.

'P(X = 2)' is read as 'the probability that the score obtained is 2'.

robability Density Function

A function which is responsible for allocating probabilities of a random variable X is called the probability density function (pdf) of X.

Pdf may be given as a table or expressed as a function of x.

umulative Distribution Function

For any real number x,

So for a given real number x, F(x) is the sum of the probabilities up to and including the events assign to x.

Properties of F(x)

• F(x) is defined for every real number x
• 0 £ F(x) £ 1
• When x ® -¥, F(x) ® 0.  When x ® ¥, F(x) ® 1
• F(x) is a non-decreasing function: if s < t, then F(s) £ F(t)
• If X is discrete, then F(x) is a step function
• P(a < X £ b) = F(b) - F(a)
   (Take note of the inequalities)

xpectation

The mean value or expectation of a drv X, written as m or E(X), is given by

Note:  If the pdf of X is symmetrical about the central value c, then E(X) = c.

In general, if g(X) is any function of the random variable X, then

Properties of E

Let a and b be any constants

• E(a) = a
• E(aX) = aE(X)
• E(aX + b) = aE(X) + b
• E[f(X) ± g(X)] = E[f(X)] ± E[g(X)]
• E(XY) = E(X)E(Y)  if X and Y are independent

ariance

The variance of a rv X, denoted by Var(X), is defined as

The standard deviation of X, denoted by s, is the square root of Var(X):

So Var(X) = s².

Computational formula for Var(X):

Properties of Var

Let a and b be constants

• Var(a) = 0
• Var(aX) = a²Var(X)
• Var(aX + b) = a²Var(X)

um of Two Random Variables

Note:   Confusions often arises over the following:

The mean value for both cases are the same as

However, the variance are different as

xample

A die, with numbers 1 to 6, is weighted such that the probability of obtaining a score is proportional to the score.  Let W denotes the score obtained when the die is tossed once.  Find the pdf, cdf, expectation and variance of W.  If X denotes three times the score when the die is tossed once, and Y denotes the sum of the scores when the die is tossed thrice, what are the expectation and variance of X and Y.

Solution:

W : score obtained when the die is tossed once
P(W = w) = kw,  for w = 1, 2, ..., 6

å P(W = w) all w	= 1
k(1 + 2 + ... + 6)	= 1
k	= 1/21

The pdf of W is  P(W = w) = w/21  for w = 1, 2, ..., 6.

For w = 1, 2, ..., 6,

F(w) =	å P(W = t) t £ w
=	(1 + ... + w)/21
=	w(w + 1)/42

The cdf of W is  F(x) = w(w + 1)/42  for w = 1, 2, ..., 6.

E(W) =	å wP(W = w) all w
=	(12 + 22 + ... + 62)/21
=	13/3

E(W2) =	å w2P(W = w) all w
=	(13 + 23 + ... + 63)/21
=	21

Var(W) =	E(W2) - E2(W)
=	21 - (13/3)2
=	20/9

X = 3W	Y = W1 + W2 + W3
E(X) = 3E(W) = 13	E(Y) = E(W1) + E(W2) + E(W3) = 13
Var(X) = 32Var(W) = 20	Var(Y) = Var(W1) + Var(W2) + Var(W3) = 20/3