A variable which can assume finite number of possible values or an infinite sequence of countable real numbers is called a discrete random variable.

**Summary**

·
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 [*t*_{1} ,*t* _{2} ] is given by the
integral of a function called the probability
density function *f* * _{X}* (

Other names that are used instead of probability
density function include density function, continuous probability function, integrating
density function.

Conditions:

*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}*
(

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 *F _{X}*
(

(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 .

(viii)

=
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** **μ_{X}** **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}*
(

·
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 ( X^{2} )−
[ E (X)]^{2}

·
If X is a random variable, the standard
deviation of X , denoted by σ_{X} , is defined as + √Var [ X ] .

·
The variance of X , denoted by σ_{X}^{2}
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) = a^{2}V ( X)

·
Raw moments

·
Central Moments

m_{r}
= E[(X–m_{X}
)^{r}]

m_{1}* *=* **E*(*X*) =* *m_{X}* *, the
mean of* **X*.

m_{1} = *E*[*X*–m* _{X}*] = 0.

m_{2} = *E*[(*X*–m* _{X}*)

Tags : Random Variable and Mathematical Expectation , 12th Business Maths and Statistics : Chapter 6 : Random Variable and Mathematical Expectation

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12th Business Maths and Statistics : Chapter 6 : Random Variable and Mathematical Expectation : Summary | Random Variable and Mathematical Expectation

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