· A variable which can assume finite number of possible values or an infinite sequence of countable real numbers is called a discrete random variable.
· Probability mass function (p.m.f.)
· Discrete distribution function (d.f.):
FX ( x) = P ( X ≤ x ) for all x ∈R
i .e., FX (x) = ∑ xi ≤ x p(xi )
· A random variable X which can take on any value (integral as well as fraction) in the interval is called continuous random variable.
· Probability density function (p.d.f.)
The probability that a random variable X takes a value in the (open or closed) interval [t1 ,t 2 ] is given by the integral of a function called the probability density function f X ( x)
Other names that are used instead of probability density function include density function, continuous probability function, integrating density function.
f ( x) ≥ 0 ∀ x
∞ ∫-∞ f ( x)dx = 1
· Continuous distribution function
If X is a continuous random variable with the probability density function f X ( x), then the function FX ( x) is defined by
is called the distribution function (d.f) or sometimes the cumulative distribution function (c.d.f) of the random variable X .
· Properties of cumulative distribution function (c.d.f.)
The function FX (X) or simply F(X) has the following properties
(iii) F(⋅) is a monotone, non-decreasing function; that is, F (a) ≤ F (b) for a < b .
(iv) F(⋅) is continuous from the right; that is, lim h → 0 F (x + h ) = F(x).
(v) F ′(x) = d/dx F (x) = f (x) ≥ 0
(vi) F ′(x) = d/dx F (x) = f (x)⇒ dF(x ) = f (x )dx
(vii) dF(x) is known as probability differential of X .
= P ( X ≤ b) − P( X ≤ a)
= F (b) − F (a)
· Mathematical Expectation
The expected value is a weighted average of the values of a random variable may assume.
· Discrete random variable with probability mass function (p.m.f.)
· Continuous random variable with probability density function
· The mean or expected value of X, denoted by μX or E(X).
· The variance is a weighted average of the squared deviations of a random variable from its mean.
· Var (X) = ∑ [x − E(X )]2 p(x)
if X is discrete random variable with probability mass function p(x).
if X is continuous random variable with probability density function f X ( x) .
· Expected value of [X –E(X)]2 is called the variance of the random variable.
i.e.,Var (X) = E [ X − E (X)]2 = E ( X2 )− [ E (X)]2
· If X is a random variable, the standard deviation of X , denoted by σX , is defined as + √Var [ X ] .
· The variance of X , denoted by σX2 or Var ( X) or V ( X) .
· Properties of Mathematical expectation
(i) E (a) = a , where ‘a’ is a constant
(ii) E (aX ) = aE ( X)
(iii) E (aX + b) = aE( X ) + b , where ‘a’ and ‘b’ are constants.
(iv) If X ≥ 0, then E(X) ≥ 0
(v) V (a) = 0
(vi) If X is random variable, then V (aX + b) = a2V ( X)
· Raw moments
· Central Moments
mr = E[(X–mX )r]
m1 = E(X) = mX , the mean of X.
m1 = E[X–mX] = 0.
m2 = E[(X–mX)2], the variance of X .