Variance of a Random Variable

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Mathematical Logic
- Statements - Introduction in Logic
- Sentences and Statement in Logic
- Truth Value of Statement
- Open Sentences in Logic
- Compound Statement in Logic
- Quantifier and Quantified Statements in Logic
- Logical Connective, Simple and Compound Statements
- Logical Connective, Simple and Compound Statements
- Examples Related to Real Life and Mathematics
- Statement Patterns and Logical Equivalence
- Algebra of Statements
- Difference Between Converse, Contrapositive, Contradiction
- Application of Logic to Switching Circuits
Mathematical Logic
- Concept of Statements
- Truth Value of Statement
- Logical Connective, Simple and Compound Statements
- Statement Patterns and Logical Equivalence
- Tautology, Contradiction, and Contingency
- Duality
- Quantifier and Quantified Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
Matrics
- Elementry Transformations
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Trigonometric Functions
- Trigonometric Equations and Their Solutions
- Solutions of Triangle
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Pair of Straight Lines
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Vectors
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Line and Plane
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Linear Programming
- Linear Inequations in Two Variables
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Matrices
- Elementary Transformations
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- Inverse of a Matrix Existance
- Adjoint Method
- Addition of Matrices
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Differentiation
- Differentiation
- Derivatives of Composite Functions - Chain Rule
- Geometrical Meaning of Derivative
- Derivatives of Inverse Functions
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- Derivatives of Implicit Functions
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- Higher Order Derivatives
Applications of Derivatives
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- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima
Indefinite Integration
- Indefinite Integration
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Definite Integration
- Definite Integral as Limit of Sum
- Fundamental Theorem of Integral Calculus
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Application of Definite Integration
- Application of Definite Integration
- Area Bounded by the Curve, Axis and Line
- Area Between Two Curves
Differential Equations
- Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Homogeneous Differential Equations
- Linear Differential Equations
- Application of Differential Equations
- Solution of a Differential Equation
Probability Distributions
- Random Variables and Its Probability Distributions
- Types of Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Distribution of a Continuous Random Variable
- Variance of a Random Variable
- Expected Value and Variance of a Random Variable
Binomial Distribution
- Bernoulli Trial
- Binomial Distribution
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)
- Bernoulli Trials and Binomial Distribution
Trigonometric Functions
- Trigonometric equations
- General Solution of Trigonometric Equation of the Type
- Solutions of Triangle
- Hero’s Formula in Trigonometric Functions
- Napier Analogues in Trigonometric Functions
- Inverse Trigonometric Functions
- Inverse Trigonometric Functions - Principal Value Branch
- Graphs of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions
Pair of Straight Lines
- Pair of Lines Passing Through Origin - Combined Equation
- Pair of Lines Passing Through Origin - Homogenous Equation
- Theorem - the Joint Equation of a Pair of Lines Passing Through Origin and Its Converse
- Acute Angle Between the Lines
- Condition for Parallel Lines
- Condition for Perpendicular Lines
- Pair of Lines Not Passing Through Origin-combined Equation of Any Two Lines
- Point of Intersection of Two Lines
Circle
- Tangent of a Circle - Equation of a Tangent at a Point to Standard Circle
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- Condition of tangency
- Tangents to a Circle from a Point Outside the Circle
- Director circle
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Conics
- Tangents and normals - equations of tangent and normal at a point
- Condition of tangency
- Tangents in terms of slope
- Tangents from a Point Outside Conics
- Locus of Points from Which Two Tangents Are Mutually Perpendicular
- Properties of Tangents and Normals to Conics
Vectors
- Vectors Revision
- Collinearity and Coplanarity of Vectors
- Linear Combination of Vectors
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- Section Formula
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- Scalar Triple Product of Vectors
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- Application of Vectors to Geometry
- Medians of a Triangle Are Concurrent
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Three Dimensional Geometry
- Introduction of Three Dimensional Geometry
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- Angle Between Two Lines
- Condition for Perpendicular Lines
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Line
- Equation of Line Passing Through Given Point and Parallel to Given Vector
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Plane
- Equation of Plane in Normal Form
- Equation of Plane Passing Through the Given Point and Perpendicular to Given Vector
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- Vector and Cartesian Equation of a Plane
- Angle Between Two Planes
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- Distance of a Point from a Plane
Linear Programming Problems
- Introduction of Linear Programming
- Mathematical Formulation of Linear Programming Problem
- Constraint Equations
- Non Negativity Restrictions
- Different Types of Linear Programming Problems
- Graphical Method of Solving Linear Programming Problems
Continuity
- Introduction of Continuity
- Continuity of a Function at a Point
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- Discontinuity of a Function
- Types of Discontinuity
- Concept of Continuity
- Algebra of Continuous Functions
- Continuity in Interval - Definition
- Exponential and Logarithmic Functions
- Continuity of Some Standard Functions - Polynomial Function
- Continuity of Some Standard Functions - Rational Function
- Continuity of Some Standard Functions - Trigonometric Function
- Continuity - Problems
Differentiation
- Revision of Derivative
- Relationship Between Continuity and Differentiability
- Every Differentiable Function is Continuous but Converse is Not True
- Derivatives of Composite Functions - Chain Rule
- Derivative of Inverse Function
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Implicit Functions
- Exponential and Logarithmic Functions
- Derivatives of Functions in Parametric Forms
- Derivative of Functions in Product of Function Form
- Derivative of Functions in Quotient of Functions Form
- Higher Order Derivative
- Second Order Derivative
Applications of Derivative
- Mean Value Theorem
- Rate of Change of Bodies or Quantities
- Increasing and Decreasing Functions
- Tangents and Normals
- Approximations
- Maxima and Minima - Introduction of Extrema and Extreme Values
- Maxima and Minima in Closed Interval
- Maxima and Minima
Integration
- Methods of Integration: Integration by Substitution
- Methods of Integration: Integration Using Partial Fractions
- Methods of Integration: Integration by Parts
- Definite Integral as the Limit of a Sum
- Fundamental Theorem of Calculus
- Properties of Definite Integrals
- Evaluation of Definite Integrals by Substitution
- Integration by Non-repeated Quadratic Factors
Applications of Definite Integral
- Area of the Region Bounded by a Curve and a Line
- Area Between Two Curves
- Volume of Solid of Revolution
Differential Equation
- Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Formation of Differential Equation by Eliminating Arbitary Constant
- Differential Equations with Variables Separable Method
- Homogeneous Differential Equations
- Linear Differential Equation
- Applications of Differential Equation
Statistics
- Bivariate Frequency Distribution
- Scatter Diagram
- Covariance of Ungrouped Data
- Covariance for Bivariate Frequency Distribution
- Karl Pearson’s Coefficient of Correlation
Probability Distribution
- Conditional Probability
- Random Variables and Its Probability Distributions
- Discrete and Continuous Random Variable
- Probability Mass Function (P.M.F.)
- Probability Distribution of a Discrete Random Variable
- Cumulative Probability Distribution of a Discrete Random Variable
- Expected Value, Variance and Standard Deviation of a Discrete Random Variable
- Probability Density Function (P.D.F.)
- Distribution Function of a Continuous Random Variable
Bernoulli Trials and Binomial Distribution
- Bernoulli Trials and Binomial Distribution
- Conditions for Binomial Distribution
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)
- Standard Deviation of Binomial Distribution (P.M.F.)
- Calculation of Probabilities
- Normal Distribution (P.D.F)

Definition

Let X be a random variable whose possible values $x_{1}, x_{2}, ..., x_{n}$ occur with probabilities $p (x_{1}), p (x_{2}), ..., p (x_{n})$ respectively.
Let µ = E (X) be the mean of X. The variance of X, denoted by Var (X) or $σ_{x}^{2}$ is defined as
$σ_{x}^{2} = V a r (X) = \sum_{i = 1}^{n} {(x_{i} - μ)}^{2} p (x_{i})$
or equivalently $σ_{x}^{2}$ = $E {(X - μ)}^{2}$
The non-negative number

$σ_{x} = \sqrt{V a r (X)} = \sqrt{\sum_{i = 1}^{n} {(x_{i} - μ)}^{2} p (x_{i})}$

is called the standard deviation of the random variable X.
Another formula to find the variance of a random variable. We know that,

Var (X) = $\sum_{i = 1}^{n} {(x_{i} - μ)}^{2} p (x_{i})$

$= \sum_{i = 1}^{n} (x_{i}^{2} + μ^{2} - 2 μ x_{i}) p (x_{i})$

$= \sum_{i = 1}^{n} x_{i}^{2} p (x_{i}) + \sum_{i = 1}^{n} μ^{2} p (x_{i}) - \sum_{i = 1}^{n} 2 μ x_{i} p (x_{i})$

$= \sum_{i = 1}^{n} x_{i}^{2} p (x_{i}) + μ^{2} \sum_{i = 1}^{n} p (x_{i}) - 2 μ \sum_{i = 1}^{n} x_{i} p (x_{i})$

$= \sum_{i = 1}^{n} x_{i}^{2} p (x_{i}) + μ^{2} - 2 μ^{2} [since \sum_{i = 1}^{n} p (x_{i}) = 1 and μ = \sum_{i = 1}^{n} x_{i} p (x_{i})]$

$= \sum_{i = 1}^{n} x_{i}^{2} p (x_{i}) - μ^{2}$

or Var (X) = $\sum_{i = 1}^{n} x_{i}^{2} p (x_{i}) - {(\sum_{i = 1}^{n} x_{i} p (x_{i}))}^{2}$

$or V a r (X) = E (x^{2}) - {[E (X)]}^{2}, where E (X^{2}) = \sum_{i = 1}^{n} x_{i}^{2} p (x_{i})$

If you would like to contribute notes or other learning material, please submit them using the button below.

x = x	-1	0	3
P (X = x)	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{2}{5}$

x = x	1	2	3
P (X = x)	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{2}{5}$

Variance of a Random Variable

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Topics

Mathematical Logic

Mathematical Logic

Matrics

Trigonometric Functions

Pair of Straight Lines

Vectors

Line and Plane

Linear Programming

Matrices

Differentiation

Applications of Derivatives

Indefinite Integration

Definite Integration

Application of Definite Integration

Differential Equations

Probability Distributions

Binomial Distribution

Trigonometric Functions

Pair of Straight Lines

Circle

Conics

Vectors

Three Dimensional Geometry

Line

Plane

Linear Programming Problems

Continuity

Differentiation

Applications of Derivative

Integration

Applications of Definite Integral

Differential Equation

Statistics

Probability Distribution

Bernoulli Trials and Binomial Distribution

Definition

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