Random Variable and Mathematical Expectation

Dear students, before you start revision for the chapter Random Variable and Mathematical Expectation, first make sure you revise all the formulas and key points. Only if you are thorough with the main concepts and formulas, you can confidently work out all the sums. That said, here are the most important formulas in Lesson 6 Random Variable and Mathematical Expectation , Class 12 Business Maths.

Important Formula in Random Variable and Mathematical Expectation

Random Variable: Discrete Random variable

Probability mass function(p.m.f)

$p(x) = \begin{cases} P(X = x_i) = p_i = p(x_i), & \text{if } x = x_i,\ i = 1, 2, \dots, n \\ 0, & \text{if } x \neq x_i \end{cases} \\$

Conditions:

$p(x_{i})\ge 0\;\forall \;i\;\text{and} \\$
$\sum_{i=1}^{\infty }p(x_{i})=1 \\$

Discrete distribution function (d.f.):

$F_{x}(x)=P(X\le x)\;\text{for all x}\;\in R \\$
$F_{x}(x)=\sum_{}^{}p(x_{i}) \\$

Probability density function (p.d.f.)

$P(t_{1}\le X\le t_{2})=\int_{t_{1}}^{t_{2}}f_{x}(x)dx \\$

Conditions:

$f(x)\ge 0\;\forall \;x \\$
$\int_{-\infty }^{\infty }f(x)=1 \\$

Continuous distribution function

$F_{x}(x)=P[X\le x]=\displaystyle \int_{-\infty }^{x }f_{x}(t)dt\; -\infty \;\lt x\lt \;\infty \\$

Properties of cumulative distribution function (c.d.f.)

$\text{The function F(X) or}\; F_{x}(x) \;\text{has the following properties:} \\$

$(i) 0\le F(x)\le 1\; -\infty \lt x\lt \infty\\$
$(ii)F(-\infty )=\lim_{x \to -\infty } F(x)=0 \;and \; F(+\infty )=\lim_{x \; \to \; \infty } F(x)=1\\$
$(iii) F(a)\le F(b)\; for\; a\lt b\\$
$(iv)\lim_{h \to 0} F(x+h)=F(x)\\$
$(v)F'(x)=\frac{d }{dx}F(x)=f(x)\ge 0\\$
$(vi)F'(x)=\frac{d }{dx}F(x)=f(x)=dF(x)=f(x)dx\\$
$(vii)dF(x)\;\text{is known as probability differential of X}\\$
$(viii)P(a \le x\le b)=\int_{b}^{a}f(x)dx=\int_{-\infty }^{b}f(x)dx-\int_{-\infty }^{a }f(x)dx\\$
$=P(X\le b)-P(X\le a)\\$
$=F(b)-F(a) \\$

Mathematical Expectation

Discrete random variable with probability mass function (p.m.f.)

$E(X)=\sum_{}^{}x P(x) \\$

Continuous random variable with probability density function

$E(X)=\sum_{-\infty }^{\infty }x F(x) dx \\$
$Var(X)=\sum_{}^{}[x-E(x)]^{2}p(x) \\$
$\text{if X is discrete random variable with probability mass function p(x).} \\$
$\text{Mean}=E(x)\\$
$\text{Variance}=E(x^{2})-[E(x)]^{2}\\$
$\text{S.D}=\sqrt{Variance (x)} \\$

Properties of Mathematical Expectation

$(i) E(a)=a\; \text{where 'a' is a constant}\\$
$(ii) E(aX)=aE(X)\\$
$(iii) E(aX+b)=aE(X)+b\; \text{where 'a' and 'b' are constants.}\\$
$(iv)\text{If X}\;\ge 0, \;\text{then}\;E(X)\ge 0\\$
$(v) V(a)= 0\\$
$(vi)\; \text{ If X is random variable, then}\; V(aX+b)=a^{2}V(X) \\$

Random Variable and Mathematical Expectation – Important Formulas

Important Formula in Random Variable and Mathematical Expectation

Mathematical Expectation

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Important Formula in Random Variable and Mathematical Expectation

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