Theoretical Distribution

Question 1

When Binomial Distribution is followed?

Answer 1

1. Trial is independent
2. The Number of trials are finite integer
3. Random variable X is followed by n and P

Question 2

f(x) = p(X=x) = Binomial Distribution formula

Answer 2

nCx * p^x * q^n-x , X = 0,1,2,…..n

Question 3

Important properties of Binomial Distribution

Answer 3

Sum of F(x) = 1

Binomial Distribution is bi parametric distribution - (n,p)
Binomial Distribution may be Uni modal or Bi Modal -

if (n+1)p is not a integer , (n+1) p
If (n+1)p is an integer, (n+1)p -1

Question 4

Poisson Distribution

Formula
Usage
e value

Answer 4

f(x) = P(x) = e^-m * m^x / X!

e = 2.71828

X = 0,1,2,3 ….. Infinity

Question 4

Formulas Binomial distribution

1. Mean
2. Variance
3. Addictive Property

Answer 4

Mean = np
SD = npq

If p and q are less than or equal to 1, npq<np

Addictive Property -

X ~ B(n,p)
Y ~ C(n,p)

(X+Y) ~ A (n+n , p+p)

Question 5

Properties of Poisson Distribution

Mean formula
Variance formula
Modal

Answer 5

1. e^-m > 0 , m>0 , f(x) Greater than equal to 0
2. uniparametric distribution m , one parametric “m”
3. mean = m
4. Variance = m
5. Integer - m and m-1
6. Non Integer - Largest integer in m

Question 6

Addictive property and
Poisson distribution Application

Answer 6

X ~ P(m1)
Y ~ P(m2)

Z = X+Y ~ P(m1+m2)

Application of Poisson Distribution:

1. No of Printing mistakes per page
2. No. of radioactive element per minute in fusion process
3. no. of Road accidents on a busy road per minute

Question 7

Normal Distribution

Formula

Answer 7

f(x) = 1/ Variance (Square root 2pi) * e^(y)

y = -(x’-u)^2 / 2 variance ^2

Question 8

Properties of Normal Distribution

QD ,Mean , MD

Answer 8

1. bi parametric , Mew and variance
2. Mean=Median=Mode = mew
3. Mean deviation = 0.8 * Variance
4. First Q = mew - 0.675 Var
5. Third Q = mew + 0675 Var
6. QD = 0.675 var
7. Normal Distribution is symmetrical about Mew = x
   its skewness is zero

Two Point of inflexion
Mew - 3Var
Mew + 3 var

99.73% - lies between Mew - 3 Var and Mew + 3 var

X ~ N(Mew , SD)
Y ~ B(Mew , SD2)

X+Y ~ A(Mew + Mew, SD + SD2)

Question 9

Standard Normal Distribution

Formula
and Properties

Answer 9

f(z) = 1/square root 2pi * e^-Z^2 /2

Mean, Mode, Median = 0
SD = 1, then MD = 0.8 and QD = 0.675
SND is symmetrical

two tails of Standard normal deviation never touch the horizontal axis