9 Jun 1999

### 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

 å P(X = x) = 1. all x

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

The cumulative distribution function (cdf) of a rv X is a function defined on R as follows:
For any real number x,
F(x) =  P(X £ x)
 å P(X = t) t £ 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

E(X) =
 å xP(X = x). all x

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

E[g(X)] =
 å g(x)P(X = x). all x

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

Var(X) = E[(X - m)2].

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

 ______ s = Ö Var(X) .

So Var(X) = s2.

Computational formula for Var(X):

Var(X) = E(X2) - [E(X)]2.

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

 If X and Y are any two random variables, then for any constants a and b,  E(aX ± bY) = aE(X) ± bE(Y).  If X and Y are also independent, then  Var(aX ± bY) = a2Var(X) + b2Var(Y).

Note:   Confusions often arises over the following:

 2X - twice the value of one observation of X X1 + X2 - the sum of two independent obsevations of X
The mean value for both cases are the same as
 E(2X) = 2E(X) E(X1 + X2) = E(X1) + E(X2) = 2E(X)
However, the variance are different as
 Var(2X) = 22Var(X) = 4Var(X) Var(X1 + X2) = Var(X1) + Var(X2) = 2Var (X).

### 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

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