Distributions

Question 1

Geometric Probability Distribution

Answer 1

When an experiment repeated until the 1ˢᵗ success follows geometric distribution.
E(x)=1/p [Expected number of trials required]
V(x)=q/p² [Variance of number of trials required]
(Here p is probability of success and q=1-p)

Question 2

Binomial Distribution
* Another Name
* ____parameter distribution
* Conditions
* X
* Formula
* Mean
* Variance
* Mode
* Probabilities are distributed in what manner
*Compare mean and variance

Answer 2

BINOMIAL DISTRIBUTION

Another name: Bernoulli Distribution

Discrete Random Variable

2 parameter distribution B(n,p)

Conditions:
* There should be finite, indpendent and small number of trials
* Exact two outome: p(success), q(fail)
* Probability should be same in each trial i.e. p remain same in each trial, q remain same in each trial

X: no. of success in n trials

Mean=np
Variance=npq
Mode=(n+1)p

In binomial distribution, probabilities are distributed for different outcomes in this manner (p+q)ⁿ

Mean > variance (np>npq)

Question 3

Types of binomial distribution

Answer 3

Symmetrical Binomial Distribution p=q=1/2
Skewed Binomial Distribution p≠q

Question 4

Poisson distribution
* ____parameter distribution and parameter called _____
* Conditions
* X
* Formula (2)
* Mean
* Variance
* Hint
* Type of Questions

Answer 4

POISSON DISTRIBUTION P(λ)

Discrete Random Variable

Uniparameter distribution P(λ) and λ is called poisson rate

Conditions:
* n is very large
* p is very small
* X is DRV, n->large and p->least so that λ =np = finite then X follows Poisson Distribution.

X: no. of events(success) happens in specific time periods or space

mean = variance = λ = np

Hint:
* Poisson word in question
* e^something given

Type of Questions:
* No. of phone calls record of any department
* No. of accidents
* No. of print mistakes in good book
* No. of death cases from very rare disease
* No. of defective material in factory

Question 5

Moment in expectation

Answer 5

E[Xʳ] pronounced as rth moment

Question 6

Mode of Poisson distribution

Answer 6

Question 7

Exponential distribution
* ____parameter distribution
* Conditions
* X
* Formula (2)
* Mean
* Variance
* Standard Deviation
* Relation between mean and standard deviation
* Type of questions

Answer 7

EXPONENTIAL DISTRIBUTION E(λ)

Continuous random variable

Uniparameter distribution E(λ)

Conditions:
* e^(something)
* answer in exponential
* RV >=0
* λ>0

X>=0

Mean = 1/λ
Variance = 1/λ²
SD= 1/λ

Mean = SD = 1/λ

Type of questions:
* Rain
* DNA
* molecules
* particles
* pressure

Question 8

rth moment formula

Answer 8

Question 9

Uniform Distribution
* Another name
* ____parameter distribution
* Conditions
* X
* Formula
* Mean
* Variance
* Standard Deviation

Answer 9

UNIFORM DISTRIBUTION U(a,b)

Rectangular Distribution

Continuous Random Variable

2 parameter distribution U(a,b)

X defined on [a,b)]

2 parameter distribution U(a,b)

Mean = (a+b)/2
Variance = (b-a)²/12
SD = (b-a)/sqrt(12)

Question 10

Standard Uniform distribution

Answer 10

In uniform distribution,
a=0, b=1
Mean = 1/2
Variance 1/12

Question 11

Graph of uniform distribution

Answer 11