Question

Sketch the p.d.f. of the beta distribution for each of the following pairs of values of the parameters:

a. α = 1/2 and β = 1/2

b. α = 1/2 and β = 1

c. α = 1/2 and β = 2

d. α = 1 and β = 1

e. α = 1 and β = 2

f. α = 2 and β = 2

g. α = 25 and β = 100

h. α = 100 and β = 25

Short answer

Sketch the PDF of each of the above pairs of beta distribution with given parameters.

Step-by-step solution

Given information

\(\alpha ,\beta \) are positive, and X is a random variable with a probability density function

\(f\left( {x;\alpha ,\beta } \right) = \frac{{\left| \!{\overline {\, {\left( {\alpha + \beta } \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( \alpha \right)} \,}} \right. \left| \!{\overline {\, {\left( \beta \right)} \,}} \right. }}{x^{\alpha - 1}}{x^{\beta - 1}}\)

Then X is said to follow the beta distribution. We need to sketch a graph for each pair of values of these parameters.

(a) Sketch of graph

Using the R programming language, we will plot the PDF of beta distribution with parameters \(\alpha = 0.5,\beta = 0.5\).

R programming Code:

curve (dbeta (x, shape1=0.5, shape2=0.5), main=expression (paste (alpha,"=0.5, “, beta,"=0.5")), ylab="Density")

(b) Sketch of graph

Using R programming language, we will plot the PDF of beta distribution with parameters\(\alpha = 0.5,\beta = 1\)

R programming Code:

curve (dbeta (x, shape1=0.5, shape2=1), main=expression (paste (alpha,"=0.5, “, beta,"=1")), ylab="Density").

(c) Sketch of graph

Using the R programming language, we will plot the PDF of beta distribution with parameters\(\alpha = 0.5,\beta = 2\).

R programming Code:

curve (dbeta (x, shape1=0.5, shape2=2), main=expression (paste (alpha,"=0.5, “, beta,"=2")), ylab="Density")

(d) Sketch of graph

Using the R programming language, we will plot the PDF of beta distribution with parameters \(\alpha = 1,\beta = 1\).

R programming Code:

curve (dbeta (x, shape1=1, shape2=1), main=expression (paste (alpha,"=1, “, beta,"=1")), ylab="Density")

(e) Sketch of graph

Using the R programming language, we will plot the PDF of beta distribution with parameters\(\alpha = 1,\beta = 2\).

R programming Code:

curve (dbeta (x, shape1=1, shape2=2), main=expression (paste (alpha,"=1, “, beta,"=2")), ylab="Density")

(f) Sketch of graph

Using the R programming language, we will plot the PDF of beta distribution with parameters\(\alpha = 2,\beta = 2\).

R programming Code:

curve (dbeta(x, shape1=2,shape2=2),main=expression(paste(alpha,"=2 , ",beta,"=2")),ylab="Density")

(g) Sketch of graph

Using the R programming language, we will plot the PDF of beta distribution with parameters\(\alpha = 25,\beta = 100\).

R programming Code:

curve (dbeta (x, shape1=25, shape2=100), main=expression (paste (alpha,"=25, “, beta,"=100")), ylab="Density")

(h) Sketch of graph

Using the R programming language, we will plot the PDF of beta distribution with parameters\(\alpha = 100,\beta = 25\).

R programming Code:

curve (dbeta (x, shape1=100, shape2=25), main=expression (paste (alpha,"=100, “, beta,"=25")), ylab="Density")