Cumulative Distribution Functions

• Oct 10th 2013, 09:26 AM
Matt1993
Cumulative Distribution Functions
I cant remember anything about CDF's Please can someone help with this question!!! :(

Obtain the cumulative distribution function of the following discrete random variables
(A) Bin(3 , theta)
(B) Unif( 0 , m)
(C) Geom( Theta )

The question did give hints but they don't help me what so ever

HINT
First calculate F at the integers. (Distinguish the index of the summation from the upper limit of summation). Secondy extend from the integers to the real line.

Any help would be appreciated!!!
• Oct 10th 2013, 10:59 PM
chiro
Re: Cumulative Distribution Functions
Hey Matt1993.

Hint: For discrete distributions P(X <= x) = Sigma (i = 0 to x) P(X = x) [Assuming that the first event is X = 0]. For continuous distributions, P(X < x) = Integral [-infinity,x] f(u)du. These are standard definitions.

Given the above, what is P(X = x) for (a) and (c) and what is f(u) for b?
• Oct 14th 2013, 07:38 AM
Matt1993
Re: Cumulative Distribution Functions
Ok so
For bin(3, Theta) then p(r) = (nCr)*(theta)^(r)*(1- theta)^(n-r)
For unif(0, m) then p(r) = 1/(m+1) for r = 0,1,2,3,...,m or 0 otherwise
For geom p(r) = (1- theta)^r * theta for 0,1,2,3,....
Were would I go from here :) thanks for this
• Oct 15th 2013, 01:27 AM
chiro
Re: Cumulative Distribution Functions
For the discrete you need to sum all values greater than or equal to a particular value in terms of probabilities.

So for Binomial (and other discrete) you have P(X <= 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3).

For uniform you have P(X < x) = Integral [0,x) f(u)du = Integral [0,x) 1*du.

Remember that for discrete you add up all individual cases that satisfy the inequality and for continuous you use the integral definition I provided above.