Let X be uniformly distributed on (0,1)

  • Thread starter TomJerry
  • Start date
Question:
Let X be uniformly distributed on (0,1). Show that Y=- [tex]\lambda[/tex]-1 1n(1-X) has an exponential distribution with parameter [tex]\lambda[/tex]>0
 

chiro

Science Advisor
4,783
127
Do you know what generating functions are?
 

statdad

Homework Helper
1,480
29
let

[tex]
F_Y(y) = P(Y \le y)
[/tex]

and use the definition of [itex] Y [/itex] to rewrite the inequality in terms of [itex] X [/itex].
 

chiro

Science Advisor
4,783
127
Here is how you can do this with a moment generating function.

Basically you can prove that a distribution is a particular one if both moment generating functions are of the same type.

So for example you've given Y = - 1/(lambda) ln(1 - X).

We are given that X is uniform on (0,1) so basically standard uniform.

The PDF of X is simply 1 with the domain (0,1).

The MGF of Y is given by M(t) = E[e^(Yt)] = Integral [0,1] 1 x e^(-{1/lambda} x ln{1-x} x t) dx = Integral [0,1] (1-x)^(-t/{lambda})

Using a substitution we get u = 1 - x, du = -dx which gives us the integral

M(t) = Integral [0,1] u^(-t/{lambda}) du.

This gives us M(t) = 1/{-t/lambda + 1} which corresponds to the MGF of the exponential distribution.
 
You can also use the standard formula for computing the density of a strictly increasing function. If X has density f_X(x) and Y=g(X) is strictly increasing, then Y has density f_Y(y)=f_X(g^-1(y))*dg^-1(y)/dy
 

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top