Location,scale, shape parameters. Link between probability distribution through parameters.

2. Presentation
Subject:
Probability distribution II
Topic:
what are shape scale and location parameters.
what is the link between probability distributions through
Parameter.
Presented to :
Dr: Afshan Riaz
Presented by:
Rakhshanda kousar
Roll no # STAT51F20S013
Bs statistics
6th (ss)

3. Statistical parameters:
In the probability distribution parameter is a value that tells
you something about population or statistical model
The most common types of parameters are
1. Location parameter
2. Scale parameter
3. Shape parameter

4. Location parameter
 This parameters decide the location of the distribution.
 It moves the curve of the distribution left to right on horizontal axis.
 Location parameters found in the families With probability density
function is to be
f(x;μ,θ)=f(x-μ;0,θ)
Where μ is the location parameter with other parameter is θ.

5. For example;

6. scale parameter
 The scale parameter compresses or stretches the entire
distribution.
 The scale parameter of a distribution determines how
much spread there is in the distribution. The larger the
scale parameter, the more spread there is in the
distribution. and vice versa.

7. For example;

8. shape parameter
 The shape parameter of a distribution effects the
general shape of the distribution.
 The larger the shape parameter, the more the
distribution tends to be skewed to the left. The
smaller the shape parameter, the more the distribution
tends to be skewed to the right

9. For example;

10. Links between probability distributions
through parameter.
 In probability theory and statistics
there are several relationships
among probability distributions.
 These relationships can be
1. One distribution is a special case of
the other
2. Transformation
3. Combinations
4. Approximations etc

11.  Weibull and exponential
distribution
 Weibull distribution with
shape parameter with k =
1 and rate parameter and
rate parameter beta is an
exponential distribution
with rate parameter beta.
 Uniform and beta
distribution
 A beta distribution with
shape parameter a=b=1 is
a continuous uniform
distribution over the real
number 0 to 1.

12.  gamma and exponential
distribution
 A gamma distribution with
shape parameter α = 1
and rate parameter β is
an exponential
distribution with rate
parameter β.
 Gamma and chi-squared
distribution
 A gamma distribution with
shape parameter α = v/2 and
rate parameter β = 1/2 is a chi-
squared distribution
with ν degrees of freedom.

13.  chi-squared and
exponential distribution
 A Student's t-distribution
with one degree of
freedom (v = 1) is a Cauchy
distribution with location
parameter x = 0 and scale
parameter γ = 1.
 Student's t-distribution
and Cauchy distribution
 A chi-squared
distribution with 2
degrees of freedom (k = 2)
is an exponential
distribution with a mean
value of 2 (rate λ = 1/2 .)

14. Uniform and exponential distribution
 The pdf of the uniform distribution is;
f(x) = 1/ (B-A) for A≤ x ≤B.
 The pdf of the exponential distribution is;
f(x) = me-mx
M=λ
By using inverse transformation
Y=ln(x)
By substitution exponential distribution linked to the uniform distribution.

15. Link between normal and log-normal
distribution;
 The normal distribution and the log-normal distribution are related to each
other through a logarithmic transformation. log-normal distribution, on the
other hand, is a continuous probability distribution of a random variable
whose logarithm is normally distributed.
 In other words, if you take the logarithm of a log-normal distributed variable,
the resulting values will follow a normal distribution.
 To establish the link between the normal and log-normal distributions,
consider a random variable X that follows a normal distribution with mean μ
and standard deviation σ. If we take the natural logarithm of X, denoted as Y
= ln(X), then Y will follow a log-normal distribution.
 By comparing the two PDFs, you can see that the only difference is the
presence of the logarithm function in the log-normal distribution.

16. Pareto and chi-square distribution;
 The link between the Pareto and chi-square distributions
can be established through the concept of order statistics.
 Consider a random sample of size n from a Pareto
distribution. If we sort the sample values in ascending
order, the kth order statistic (X(k)) will follow a Pareto
distribution with parameters α and θ/k.
 Now, if we square the kth order statistic (X(k))^2, it will
follow a chi-square distribution with ν = 2k degrees of
freedom. This relationship holds true for any k in the
range 1 to n.

17. t-distribution and the F-distribution
 The link between the t-distribution and the F-distribution
arises when considering the test statistic used in the
analysis of variance.
 In ANOVA, the test statistic follows an F-distribution.
However, when there are only two groups being
compared, the F-distribution reduces to a t-distribution.
This is because the square of a t-distributed random
variable with a certain number of degrees of freedom
follows an F-distribution with the numerator degrees of
freedom equal to 1.