Topics
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
- Overview of Mathematical Logic
Matrices
- Elementry Transformations
- Properties of Matrix Multiplication
- Application of Matrices
- Applications of Determinants and Matrices
- Overview of Matrices
Trigonometric Functions
- Trigonometric Equations and Their Solutions
- Solutions of Triangle
- Inverse Trigonometric Functions
- Overview of Trigonometric Functions
Pair of Straight Lines
- Combined Equation of a Pair Lines
- Homogeneous Equation of Degree Two
- Angle between lines represented by ax2 + 2hxy + by2 = 0
- General Second Degree Equation in x and y
- Equation of a Line in Space
- Overview of Pair of Straight Lines
Vectors
Line and Plane
- Vector and Cartesian Equations of a Line
- Distance of a Point from a Line
- Distance Between Skew Lines and Parallel Lines
- Equation of a Plane
- Angle Between Planes
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Overview of Line and Plane
Linear Programming
Differentiation
- Differentiation
- Derivatives of Composite Functions - Chain Rule
- Geometrical Meaning of Derivative
- Derivatives of Inverse Functions
- Logarithmic Differentiation
- Derivatives of Implicit Functions
- Derivatives of Parametric Functions
- Higher Order Derivatives
- Overview of Differentiation
Applications of Derivatives
- Applications of Derivatives in Geometry
- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima
- Overview of Applications of Derivatives
Indefinite Integration
Definite Integration
- Definite Integral as Limit of Sum
- Integral Calculus
- Methods of Evaluation and Properties of Definite Integral
- Overview of Definite Integration
Application of Definite Integration
- Application of Definite Integration
- Area Bounded by the Curve, Axis and Line
- Area Between Two Curves
- Overview of Application of Definite Integration
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
- Overview of Differential Equations
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
- Overview of Probability Distributions
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
- Overview of Binomial Distribution
Text
Bernoulli trials:
The outcome of any trial is independent of the outcome of any other trial. In each of such trials, the probability of success or failure remains constant. Such independent trials which have only two outcomes usually referred as ‘success’ or ‘failure’ are called Bernoulli trials.
Definition:
Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions :
(i) There should be a finite number of trials.
(ii) The trials should be independent.
(iii) Each trial has exactly two outcomes : success or failure.
(iv) The probability of success remains the same in each trial.
For example, throwing a die 50 times is a case of 50 Bernoulli trials, in which each trial results in success (say an even number) or failure (an odd number) and the probability of success (p) is same for all 50 throws. Obviously, the successive throws of the die are independent experiments. If the die is fair and have six numbers 1 to 6 written on six faces, then
p = `1/2 and q = 1 -p =1/2` probability of failure.
Binomial distribution:
Let us take the experiment made up of three Bernoulli trials with probabilities p and q = 1 – p for success and failure respectively in each trial. The sample space of the experiment is the set
S = {SSS, SSF, SFS, FSS, SFF, FSF, FFS, FFF}
The number of successes is a random variable X and can take values 0, 1, 2, or 3. The probability distribution of the number of successes is as below :
P(X = 0) = P(no success)
= P({FFF}) = P(F) P(F) P(F)
= `q . q . q = q^3` since the trials are independent
P(X = 1) = P(one successes)
= P({SFF, FSF, FFS})
= P({SFF}) + P({FSF}) + P({FFS})
= P(S) P(F) P(F) + P(F) P(S) P(F) + P(F) P(F) P(S)
= `p.q.q + q.p.q + q.q.p = 3pq^2 `
P(X = 2) = P (two successes)
= P({SSF, SFS, FSS})
= P({SSF}) + P ({SFS}) + P({FSS})
= P(S) P(S) P(F) + P(S) P(F) P(S) + P(F) P(S) P(S)
= `p.p.q. + p.q.p + q.p.p = 3p^2q`
and
P(X = 3) = P(three success) = P ({SSS})
= P(S) . P(S) . P(S) = `p^3`
Thus, the probability distribution of X is
| X | o | 1 | 2 | 3 |
| P(X) | `q^3` | `3q^2p` | `3qp^2` | `p^3` |
Also, the binominal expansion of `(q + p)^3` is
`q^3 + 3q^2p + 3 qp^2 + p^3`
Thus, we may conclude that in an experiment of n-Bernoulli trials, the probabilities of 0, 1, 2,..., n successes can be obtained as 1st, 2nd,...,`(n + 1)^(th)` terms in the expansion of `(q + p)^n`.
The probability of x successes in n-Bernoulli trials is `(n!)/ (x!(n - x)!) p^x q^(n -x)`
or `"^nC_x p^x q^(n-x)`
The probability of x successes P(X = x) is also denoted by P(x) and is given by
P(x) = `"^nC_x q^(n-x) p^x` , x = 0, 1,..., n. (q = 1 – p)
This P(x) is called the probability function of the binomial distribution.
