The document provides a comprehensive overview of statistical inference, covering topics such as probability density functions, moment generating functions, sampling distributions, and the gamma and beta distributions. It explains key concepts such as degrees of freedom, chi-square, and exponential distributions, with formulas for probability density functions and moment generating functions for each distribution. Additionally, it emphasizes the practical applications of these statistical concepts in various fields.

1. D
BASIC OF STATISTICAL INFERENCE PART-II
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2. D
TYPES OF SAMPLING DISTRIBUTION
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3. KEY CONCEPTS FOR SAMPLING DISTRIBUTION
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PROBABILITY DENSITY FUNCTION
Probability density function (PDF), in statistics, a function whose integral is calculated
to find probabilities associated with a continuous random variable
(see continuity; probability theory). Its graph is a curve above the horizontal axis that
defines a total area, between itself and the axis, of 1. The percentage of this area
included between any two values coincides with the probability that the outcome of
an observation described by the probability density function falls between those
values. Every random variable is associated with a probability density function (e.g., a
variable with a normal distribution is described by a bell curve).If X be continuous
Random variable taking any continuous real values then f(x) is a probability density
function if:-
i. 𝑓 𝑥 ≥ 0 ∀𝑥
ii. −∞
∞
𝑓 𝑥 𝑑𝑥 = 1
MOMENT GENERATING FUCTION
The moment generating function (m.g.f.) of a random variable X (about origin) having
the probability function 𝑓(𝑥) is given by:
𝑀 𝑋 𝑡 = 𝐸 𝑒 𝑡𝑥 =
𝑒 𝑡𝑥 𝑓 𝑥 𝑑𝑥, (𝑓𝑜𝑟 𝑐𝑜𝑛𝑡𝑖𝑛𝑖𝑜𝑢𝑠 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛)
𝑥
𝑒 𝑡𝑥 𝑓 𝑥 , (𝑓𝑜𝑟 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛)
The integration of summation being extended to the entire range of x, t being the
real parameter and it is being assumed that the right-hand side of the equation is
absolutely convergent for some positive number h such that –h<t<h.
SAMPLING DISTRIBUTION
It may be defined as the probability law which the statistic follows , if repeated
random samples of a fixed size are drawn from specified population. A number of
samples, each of size n, are taken from the same population and if for each sample
the values of the statistic is calculated, a series of values of the statistic will be
obtained. If the number of samples is large, these may be arranged into a frequency
table. The frequency distribution of the statistic that would be obtained if the
number of samples, each of the same size (say n), were infinite is called the Sampling
distribution of the statistic
DEGREES OF FREEDOM
The term degrees of freedom (df) refers to the number of independent sample
points used to compute a statistic minus the number of parameters estimated from
the sample points: For example, consider the sample estimate of the population
variance 𝑠2
𝑠2 =
𝑖=1
𝑛
𝑋𝑖 − 𝑋 2
(𝑛 − 1)
Where 𝑋𝑖 is the score for observation i in the sample, 𝑋 is the sample estimate of the
population mean, n is the number of observation in the sample. The formula is based
on n independent sample points and one estimated population parameter ( 𝑥).
Therefore, the number of degrees of freedom is n minus one. For this example
df=n-1

4. 4
GAMMA FUNCTION & BETA FUNCTION
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GAMMA FUNCTION BETA FUNCTION
The Gamma function is defined for x>0 in integral form by the improper integral
known as Euler’s integral of the second kind.
Γ 𝑥 =
0
∞
𝑡 𝑥−1 𝑒−𝑡 𝑑𝑡
Many probability distributions are defined by using the gamma function, such as
gamma distribution, beta distribution, chi-squared distribution, student’s t-
distribution etc. For data scientists, machine learning engineers, researches, the
Gamma function is probably one of the most widely used function because it is
employed in many distributions.
The Beta function is a function of two variables that is often found in probability
theory and mathematical statistics.
The Beta function is a function 𝐵: ℝ++
2
→ ℝ defined as follows:
𝐵 𝑥, 𝑦 =
Γ(𝑥)Γ(𝑦)
Γ(𝑥 + 𝑦)
There is also a Euler’s integral of the first kind.
For example, as a normalizing constant in the probability density functions of the F
distribution and of the Student’s t distribution
RELATION BETWEEN GAMMA AND BETA FUNCTION
In the realm of Calculus, many complex integrals can be reduced to expressions involving the Beta Function. The Beta Function is important in calculus due to its close connection
to the Gamma Function which is itself a generalization of the factorial function.
We know,
Γ 𝑥 =
0
∞
𝑡 𝑥−1 𝑒−𝑡 𝑑𝑡
So, the product of two factorials as
Γ 𝑢 Γ 𝑣 =
0
∞
𝑡 𝑢−1 𝑒−𝑡 𝑑𝑡
0
∞
𝑡 𝑣−1 𝑒−𝑡 𝑑𝑠 =
0
∞
0
∞
𝑒−(𝑡+𝑠) 𝑡 𝑢−1 𝑠 𝑣−1 𝑑𝑡 𝑑𝑠
Now apply the changes of variables 𝑡 = 𝑥𝑦 and 𝑠 = 𝑥(1 − 𝑦) to this double integral. Note that 𝑡 + 𝑠 = 𝑥 and that 0 < 𝑡 < ∞ and 0 < 𝑥 < ∞ and 0 < 𝑦 < 1. The jacobian of this
transformation is
𝜕(𝑡,𝑠)
𝜕(𝑥,𝑦)
=
𝑦 𝑥
1 − 𝑦 −𝑥 = −𝑥𝑦 − 𝑥 + 𝑥𝑦 = −𝑥
Since 𝑥 > 0 we conclude that 𝑑𝑡 𝑑𝑠 =
𝜕(𝑡,𝑠)
𝜕(𝑥,𝑦)
𝑑𝑥 𝑑𝑦 = 𝑥 𝑑𝑥 𝑑𝑦. Hence we have
Γ 𝑢 Γ 𝑣 =
0
1
0
∞
𝑒−𝑥 𝑥 𝑢−1 𝑦 𝑢−1 𝑥 𝑣−1 1 − 𝑦 𝑣−1 𝑥 𝑑𝑥 𝑑𝑦 =
0
∞
𝑒−𝑥 𝑥 𝑢+𝑣−1 𝑑𝑥
0
1
𝑦 𝑢−1 1 − 𝑦 𝑣−1 𝑑𝑦 = Γ 𝑢 + 𝑣 𝐵 𝑢, 𝑣
Therefore,
𝐵 𝑥, 𝑦 =
Γ(𝑥)Γ(𝑦)
Γ(𝑥 + 𝑦)

5. 5
GAMMA DISTRIBUTION
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GAMMA DISTRIBUTION
The gamma distribution is a widely used distribution. It is a right-skewed probability distribution. These distribution are useful in real life where something has a natural
minimum of 0.
If X be a continuous random variable taking only positive values, then X is said to be following a gamma distribution iff its p.d.f can be expressed as:- 𝑓 𝑥 =
λ 𝑣.𝑒−λ𝑥.𝑥 𝑣−1
Γ(𝑣)
. 𝑥 > 0
= 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 … . (1)
 Probability Density Function for Gamma Distribution
For (1), to be the Probability Density Function, we must have:-
(i) 𝑓 𝑥 ≥ 0 ∀ 𝑥 > 0 and (ii) −∞
∞
𝑓 𝑥 𝑑𝑥 = 1
Now, 𝑓 𝑥 > 0 if 𝑥 > 0 & 𝑓 𝑥 = 0 if x taking any non-positive values, so, 𝑓 𝑥 ≥ 0 ∀𝑥.
Hence, condition (i) is satisfied. Now,
−−∞
∞
𝑓 𝑥 𝑑𝑥 = −∞
0
𝑓 𝑥 𝑑𝑥 + 0
∞
𝑓 𝑥 𝑑𝑥 = 0
∞
𝑓 𝑥 𝑑𝑥 ∴ −∞
0
𝑓 𝑥 = 0 𝑎𝑠 𝑓 𝑥 = 0
Let, 𝐼 = 0
∞ λ 𝑣.𝑒λ.𝑥.𝑥 𝑣−1
┌ 𝑣
𝑑𝑥 =
λ 𝑣
┌ 𝑣 0
∞
𝑒−λ.𝑥. 𝑥 𝑣−1 … (2)
Let, λ. 𝑥 = 𝑧 , 𝑜𝑟, 𝑥 =
𝑧
λ
… … 3 Also, 𝑜𝑟, 𝑑𝑥 =
1
λ
𝑑𝑧 … . . 4
Using (3) & (4) in (2) we get:-
𝐼 =
λ 𝑣
┌(𝑣) 0
∞
𝑒−𝑧 . 𝑧 𝑣−1 1
λ
𝑣−1
.
1
λ
𝑑𝑧 =
λ 𝑣
┌ 𝑣
.
1
λ 𝑣 . 0
∞
𝑒−𝑧 . 𝑧 𝑣−1 𝑑𝑧 =
λ 𝑣
┌ 𝑣
.
┌(𝑣)
λ 𝑣 = 1
Hence, 𝑓(𝑥) statistics condition (ii).
So, equation (1) is a proper pdf.
 Moment Generating Function for Gamma Distribution
Moment generating functions are general procedure of finding out moments of a probability distribution mathematically it may be express as-𝑀 𝑥 𝑡 = 𝐸 𝑒 𝑥𝑡
This represents raw moments of the random variable X about to the origin 0.
Three important properties of m.g.f. are:- (i) 𝑀 𝑥𝑒 𝑡 = 𝑀 𝑋 𝑐𝑡 ; where c is a constant. (ii) If 𝑋𝑖’s are independent Random variables i.e. 𝐿 = 1,2, … , 𝑛; then 𝑀 𝑖=1
𝑛
𝑥 𝑖
𝑡 =
𝑖=1
𝑛
𝑀 𝑥𝑖 𝑡 . (iii) If X and Y are two random variables and if 𝑀 𝑥 𝑡 = 𝑀 𝑦(𝑡) then X and Y are two identical distribution this is called the uniqueness property, Calculating the
m.g.f. of gamma distribution:
𝑀 𝑥 𝑡 = 𝐸 𝑒 𝑥𝑡 = 0
∞
𝑒 𝑥𝑡. λ 𝑣. 𝑒−λ.𝑥.
𝑥 𝑣−1
┌(𝑣)
𝑑𝑥 =
λ 𝑣
┌ 𝑣 0
∞
𝑒 𝑥𝑡 . 𝑒λ.𝑥. 𝑥 𝑣−1 𝑑𝑥 =
λ 𝑣
┌(𝑣) 0
∞
𝑒−𝑥 λ−𝑡 . 𝑥 𝑣−1 𝑑𝑥. . . (1)
𝒑𝒖𝒕 𝒙 𝝀 − 𝒕 = 𝒛 𝑜𝑟, 𝑥 =
𝑧
λ − 𝑡
… 2 𝑎𝑙𝑠𝑜; 𝑑𝑥 λ − 𝑡 = 𝑑𝑧 𝑜𝑟, 𝑑𝑥 =
𝑑𝑧
λ − 𝑡
… (3)
Using (2) and (3) in (1) we get: 𝑀 𝑥 𝑡 =
λ 𝑣
┌(𝑣) 0
∞
𝑒−𝑧.
𝑧
λ−𝑡
𝑣−1
.
𝑑𝑧
λ−𝑡
=
λ 𝑣
┌ 𝑣 0
∞
𝑒−𝑧 . 𝑧 𝑣−1. 𝑑𝑧
1
λ−𝑡
𝑣
.
1
λ−𝑡 −1 .
1
λ−𝑡
=
λ 𝑣
┌ 𝑣
. ┌ 𝑣 .
1
λ−𝑡
𝑣
. λ − 𝑡 .
1
λ−𝑡
=
λ
λ−𝑡
𝑣
∴Required m.g.f =
1
1− 𝑡
λ
𝑣
= 1 − 𝑡
λ
−𝑣
X
Y
𝜶 = 𝟑, 𝜷 = 𝟎. 𝟓
𝜶 = 𝟑, 𝜷 = 𝟏
𝜶 = 𝟐, 𝜷 = 𝟏
𝜶 = 𝟏, 𝜷 = 𝟏

6. 6
BETA DISTRIBUTION
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BETA DISTIBUTION
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape
parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution.
The beta distribution has been applied to model the behaviour of random variables limited to intervals of finite length in a wide variety of disciplines.
 Probability Density Function for Beta Distribution
The probability density function (PDF) of the beta distribution, for 0 ≤ x ≤ 1, and shape parameters α, β > 0, is a power function of the variable x and of its reflection (1 − x) as
follows:
𝒇 𝒙; 𝜶, 𝜷 = 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕. 𝒙 𝜶−𝟏(𝟏 − 𝒙) 𝜷−𝟏
=
𝒙 𝜶−𝟏(𝟏 − 𝒙) 𝜷−𝟏
𝟎
𝟏
𝒖 𝜶−𝟏(𝟏 − 𝒖) 𝜷−𝟏 𝒅𝒖
=
Ґ(𝜶 + 𝜷)
Ґ 𝜶 Ґ(𝜷)
𝒙 𝜶−𝟏(𝟏 − 𝒙) 𝜷−𝟏=
𝟏
𝑩(𝜶, 𝜷)
𝒙 𝜶−𝟏(𝟏 − 𝒙) 𝜷−𝟏
Where Γ(z) is the gamma function. The beta function, B, =+is a normalization constant to ensure
that the total probability integrates to 1. In the above equations x is a realization—an observed
value that actually occurred—of a random process X.
This definition includes both ends x = 0 and x = 1, which is consistent with the definitions for other
continuous distributions supported on a bounded interval which are special cases of the beta
distribution, for example the arcsine distribution, and consistent with several authors, like
N. L. Johnson and S. Kotz. However, the inclusion of x = 0 and x = 1 does not work for α, β < 1;
accordingly, several other authors, including W. Feller, choose to exclude the ends x = 0 and
x = 1, (so that the two ends are not actually part of the domain of the density function) and
consider instead 0 < x < 1. Several authors, including N. L. Johnson and S. Kotz, use the
symbols p and q (instead of α and β) for the shape parameters of the beta distribution,
reminiscent of the symbols are traditionally used for the parameters of the Bernoulli distribution,
because the beta distribution approaches the Bernoulli distribution in the limit when both
shape parameters α and β approach the value of zero.
In the following, a random variable X beta-distributed with parameters α and β will be denoted by:
𝑋~𝐵𝑒𝑡𝑎(𝛼, 𝛽)
Other notations for beta-distributed random variables used in the statistical literature are 𝑋 − 𝐵𝑒 𝛼, 𝛽 𝑎𝑛𝑑 𝑋~𝛽 𝛼,𝛽.
 Moment Generating Function for Beta Distribution
The moment generating function 𝑀 𝑋 of X is given by: 𝑀 𝑋 𝑡 = 1 + 𝑘=1
∞
𝑟=0
𝑘−1 𝛼+𝑟
𝛼+𝛽+𝑟
𝑡 𝑘
𝑘!
From the definition of a moment generating function: 𝑀 𝑋 𝑡 = 𝐸 𝑒 𝑡𝑥 = 0
1
𝑒 𝑡𝑥 𝑓𝑋(𝑥) . 𝑑𝑥
So: 𝑀 𝑋 𝑡 =
1
𝐵 𝛼,𝛽 0
1
𝑒 𝑡𝑥 𝑥 𝛼−1 1 − 𝑥 𝛽−1 𝑑𝑥 =
1
𝐵 𝛼,𝛽 0
1
𝑘=0
∞ 𝑡𝑥 𝑘
𝑘!
𝑥 𝛼−1 1 − 𝑥 𝛽−1 𝑑𝑥
=
𝑘=1
∞
𝑡 𝑘
𝑘!
𝐵 𝛼 + 𝐾, 𝛽
𝐵 𝛼, 𝛽
=
𝐵(𝛼, 𝛽)𝑡0
𝐵 𝛼, 𝛽 0!
+
𝑘=1
∞
𝑡 𝑘
𝑘!
𝐵 𝛼 + 𝑘, 𝛽
𝐵 𝛼, 𝛽
= 1 +
𝑘=1
∞
𝛤(∞) 𝑟=0
𝑘
(𝛼 + 𝑟)
𝛤(𝛼)
.
𝛤 𝛼 + 𝛽
𝛤 𝛼 + 𝛽 𝑟=0
𝑘
𝛼 + 𝛽 + 𝑟
𝑡 𝑘
𝑘!
= 1 +
𝑘=1
∞
𝑟=0
𝑘−1 𝛼 + 𝑟
𝛼 + 𝛽 + 𝑟
𝑡 𝑘
𝑘!

7. 7
CHI-SQUARE AND EXPONENTIAL DISTRIBUTION
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CHI-SQUARE DISTRIBUTION EXPONENTIAL DISTRIBUTION
A chi-square distribution is defined as the sum of the squares of standard normal variates. Let x
be a random variable which follows normal distribution with mean 𝜇 & variance 𝜎2
,
𝑋~𝑁 𝜇, 𝜎2
. then standard normal variate is defined as: - 𝑍𝑖 =
𝑥 𝑖−𝜇
𝜎
The variate Z is a said to follow a standard normal distribution with mean 0 and variance 1. Let X
be a random variable containing observations, 𝑥1, 𝑥2, … , 𝑥 𝑛 .
Then the chi-square distribution is defined as:- 𝑖=1
𝑛
𝑧𝑖
2
= 𝑖=1
𝑛 𝑥 𝑖−𝜇
𝜎
2
So we can say:-
χ2
=
𝑖=1
𝑛
𝑥𝑖 − 𝜇𝑖
𝜎𝑖
2
A chi-square distribution with ‘n’ degree of freedom, where degrees of freedom refer to number
of independent association among variables.
 The Probability Density Function of a Chi-Square Distribution:
𝑓 𝑥 =
1
2
𝑛
2
. 𝑒−1
2 𝑥
. 𝑥
𝑛
2−1
┌
𝑛
2
; 𝑥 > 0
= 0 ; 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
 The Moment Generating Function of Chi-Square Distribution:
𝑀 𝑋 𝑡 = 𝐸 𝑒 𝑥𝑡
𝑀 𝑋 𝑡 =
0
∞
𝑒 𝑥𝑡
.
1
2
𝑛
2
. 𝑒−1
2 𝑥
. 𝑥
𝑛
2−1
┌ 𝑛
2
𝑑𝑥 =
0
∞
1
2
𝑛
2
. 𝑒 𝑥𝑡
. 𝑒−1
2 𝑥
. 𝑥
𝑛
2−1
𝑑𝑥
┌ 𝑛
2
=
1
2
𝑛
2
┌ 𝑛
2 0
∞
𝑒 𝑡−1
2 𝑥
. 𝑥
𝑛
2−1
𝑑𝑥 =
1
2
𝑛
2
┌ 𝑛
2 0
∞
𝑒− 1
2−𝑡 𝑥
. 𝑥
𝑛
2−1
𝑑𝑥 … . . 1
Let, 𝑥
1
2
− 𝑡 = 𝑧 𝑜𝑟, 𝑥 =
𝑧
1
2
−𝑡
… . (2) Also,
1
2
− 𝑡 𝑑𝑥 = 𝑑𝑧 𝑜𝑟, 𝑑𝑥 =
𝑑𝑧
1
2
−𝑡
… . . (3)
Using (2) and (3) we get:- 𝑀 𝑥 𝑡 =
1
2
𝑛
2
┌ 𝑛
2 0
∞
𝑒−𝑧
. 𝑧
𝑛
2−1
.
1
1
2−𝑡
𝑛
2
−1
𝑑𝑧.
1
1
2
−𝑡
=
1
2
𝑛
2
┌ 𝑛
2 0
∞
𝑒−𝑧
. 𝑧
𝑛
2−1
.
1
1
2
−𝑡
𝑛
2−1
.
1
1
2
−𝑡
𝑑𝑧
=
1
2
𝑛
2
┌
𝑛
2
┌
𝑛
2
1
1
2
−𝑡
𝑛
2
.
1
2−𝑡
1
2−𝑡
=
1
2
1
2
−𝑡
𝑛
2
=
1
1− 𝑡
1
2
𝑛
2
𝑠𝑜, 𝑀 𝑋 𝑡 = 1 −
𝑡
1
2
𝑛
2
= 1 − 2𝑡 − 𝑛
2 … (4)
(4) is a required m.g.f. of the chi-square distribution.
The Exponential distribution is one of the widely used continuous distributions. It is often used to
model the time elapsed between events.
 The Probability Density Function of Exponential Distribution:
Let X be a continuous random variable assuming only real values then X is said to be following an
exponential distribution iff:-
𝑓 𝑥 = λ. 𝑒−λ.𝑥
, 𝑥 > 0
= 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Therefore, exponential distribution is a special case of gamma
distribution with v = 1.
 The Moment Generating Function of Exponential Distribution:
Let 𝑋~𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 λ , we can find its expected value as follows, using integration by parts:
𝑀 𝑥 𝑡 = 𝐸 𝑒 𝑡𝑥
= 𝜃
0
∞
𝑒 𝑡𝑥
𝑒−λ𝑥
= 𝜃
0
∞
𝑒𝑥𝑝 − 𝜃 − 𝑡 𝑥 𝑑𝑥
=
λ
λ − 𝑡
= 1 −
𝑡
λ
−1
=
𝑟=0
∞
𝑡
λ
𝑟
, 𝜃 > 𝑡
∴ 𝑢 𝑟
′
= 𝐸 𝑋 𝑟
= 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓
𝑡 𝑟
𝑟!
𝑖𝑛 𝑀 𝑥 𝑡 =
𝑟!
𝜃 𝑟
; 𝑟 = 1,2, … .
𝑀𝑒𝑎𝑛 = 𝜇1
′
=
1
λ
Now let’s find 𝑉𝑎𝑟 𝑋 , we have
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝜇2 = 𝜇′2 − 𝜇1
′2
=
2
λ2
−
1
λ2
=
1
λ2
Thus, we obtain
If 𝑋~𝑒𝑥𝑝 λ , then Mean =
1
λ
and Variance =
1
λ2.
λ=0.5
λ=1
λ=1.5

8. 8
T-DISTRIBUTION & F-DISTRIBUTION
Copyright © 2020, DexLab Solutions Corporation
F-DISTRIBUTIONT-DISTRIBUTION
 Student’s t-distribution:
If 𝑥1, 𝑥2, … , 𝑥 𝑛 be ‘n’ random samples drawn from a normal population having mean
𝜇 & standard deviation 𝜎, then the statistic 𝑡 =
𝑥−𝜇
𝑠
𝑛
is following student t-distribution
with (n-1) degrees of freedom.
Where, 𝑥 =
1
𝑛 𝑖=1
𝑛
𝑥𝑖 ; 𝑠2 =
1
(𝑛−1) 𝑖=1
𝑛−1
𝑥𝑖 − 𝑥 𝑠
𝑛
→ standard error.
 Fisher’s t-distribution:
Let 𝑋~𝑁(0,1) & let the random variable 𝑌~𝑋 𝑛
2. Both X & Y are independent random
variables. Then the fisher’s t-distribution is define as :-
𝑡 =
𝑋
𝑌
𝑛
~𝑡 𝑛
 Probability Density Function for t-distribution:
𝑓 𝑡 =
1
𝑣𝛽
1
2
,
𝑣
2
.
1
1 + 𝑡2
𝑣
. 𝑣 + 1
2
Where, 𝑡2 > 0 Where, 𝑣 = 𝑛 − 1 degrees of freedom
= 0 , otherwise
For Fisher’s t-distribution:
𝑓 𝑡 =
1
𝑛𝛽
1
2
,
𝑛
2
.
1
1 + 𝑡2
𝑛
𝑛+1
2
Where, 𝑡2 > 0
= 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
 Application of t-distribution:
If 𝒙 𝟏, 𝒙 𝟐, 𝒂𝒏𝒅 𝒙 𝟑 are independent random variable. Each following a standard
normal distribution. What will be the distribution of
𝟐𝒙 𝟏
𝒙 𝟐
𝟐+𝒙 𝟑
𝟐
Answer:- According to the problem, we have:- 𝑥1~𝑁(0,1), 𝑥2~𝑁(0,1), 𝑥3~𝑁(0,1)
Let the statistic be ‘T’ so that: 𝑇 =
2𝑥1
𝑥2
2+𝑥3
2
𝑜𝑟, 𝑇 =
𝑥1
𝑥2
2+𝑥3
2
2
𝑜𝑟, 𝑇 =
𝑥1
𝑥2
2+𝑥3
2
2
~
𝑁 0,1
𝑥 𝑛
2
𝑛
=
𝑡 𝑛
So, 𝑇 =
2𝑥1
𝑥2
2+𝑥3
2
~𝑡𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ ′𝑛′𝑑𝑒𝑔𝑟𝑒𝑒𝑜𝑓 𝑓𝑟𝑒𝑒𝑑𝑜𝑚
The F-Distribution is a ration of two chi-square distributions. If X be a random variable
which follows a fisher’s t-distribution. Then:-𝑋~
𝑁 0,1
𝑋 𝑛
2
𝑛
Squaring the above expression we get:
𝑋2~
(𝑁 0,1 )2
𝑙𝑎𝑚𝑑𝑎 𝑛
2
𝑛
𝑜𝑟, 𝑋2~
𝑋1
2
1
𝑋 𝑛
2
𝑛
The R.V. 𝑋2~𝐹1, 𝑛. Then we say 𝑋2 follows F-distribution with 1,n degrees of freedom.
 Probability Density Function for F-distribution:
𝑓 𝐹 =
𝑣1
𝑣2
𝑣1
𝑣2
−1
. 𝐹
𝑣1
𝑣2
−1
𝛽
𝑣1
2
,
𝑣2
2
1 +
𝐹
𝑣1
𝑣1+𝑣2
2
When 𝐹 > 0.
 Application of F-Distribution:
Let 𝒙 𝟏, 𝒙 𝟐, … , 𝒙 𝒏 be a random sample drawn from a normal population with mean
𝝁 & variance 𝝈 𝟐. where both 𝝁 & 𝜎 are unknown. Obtain the 𝑴𝑳𝑬𝒔 𝒐𝒇 𝜽.
Let 𝑥1, 𝑥2, … , 𝑥 𝑛 be ‘n’ random samples drawn from a manual population with mean
𝜇 & variance 𝜎2. The p.d.f. of the normal distribution is 𝑓 𝜇, 𝜎 =
1
𝜎 2𝜋
. 𝑒
−1
2
𝑥−𝜇
𝜎
2
.
The likelihood function for the normal distribution is: 𝐿 𝜇, 𝜎 = 𝑖=1
𝑛
𝑓 𝜇, 𝜎 =
1
2𝜋𝜎
𝑛
. 𝑒−
1
2
𝜎2
𝑖=1
𝑛
𝑥 𝑖−𝜇
Taking logarithms on both sides; we get:-
𝐿𝑜𝑔 𝐿 𝜇, 𝜎 = 𝑙𝑜𝑔
1
2𝜋𝜎
−𝑛
+ 𝑙𝑜𝑔𝑒−
1
2
𝜎2
𝑖=1
𝑛
𝑥 𝑖−𝜇
𝑜𝑟, 𝐿𝑜𝑔 𝐿 𝜇, 𝜎
= −𝑛 log 𝑒 𝜎 − 𝑛 log 𝑒 2𝜋 −
1
2𝜎2
𝑖=1
𝑛
𝑥𝑖 − 𝜇
For estimator of 𝜇 & 𝜎: 𝑡ℎ𝑒 𝐹𝑜𝐶𝑠 𝑎𝑟𝑒: −
𝜕 log 𝑒 𝐿 𝜇,𝜎2
𝜕𝜇
= 0,
𝜕 log 𝑒 𝐿 𝜇,𝜎
𝜕𝜎
= 0
P(X)
X
v=+∞
V=5
V=2
V=1
V1=5,v2=2
V1=100,v2=100
V1=10,v2=1

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