Probability density function

[Macky15]

Hi. Can somebody help me to solve this? Ive been trying to figure this for a week. I'm still confuse if this will need of integrals.


For each of the following, find the constant c so that p(x) satisfies the condition of being a probability density function of a random variable X:




I. p(x) = c(2/3)^x, x ∈ N
II. p(x) = cx, x ∈ {1,2,3,4,5,6}

Hints are very much appreciated. Thanks.

 

[HallsofIvy]

First, because x is integer valued, no there is no "integral" involved, just a sum over, in the first case, all integers, and, in the second case, over just x= 1, 2, 3, 4, 5, 6.

For (I) you must have \(\displaystyle \sum_{x= 1}^\infty c(2/3)^x= c\sum_{x= 1}^\infty (2/3)^x= 1\) Do you see that that sum is a geometric series? Do you know a formula for that? (Be careful, your formula may be for 0 to infinity, while this is for 1 to infinity. If your formula does start at 0, just subtract off the the "0" term which is 1.

For (II) you just have the finite sum \(\displaystyle \sum_{x=1}^6 cx= c+ 2c+ 3c+ 4c+ 5c+ 6c= 1\).

 

[Macky15]

First, because x is integer valued, no there is no "integral" involved, just a sum over, in the first case, all integers, and, in the second case, over just x= 1, 2, 3, 4, 5, 6.

For (I) you must have \(\displaystyle \sum_{x= 1}^\infty c(2/3)^x= c\sum_{x= 1}^\infty (2/3)^x= 1\) Do you see that that sum is a geometric series? Do you know a formula for that? (Be careful, your formula may be for 0 to infinity, while this is for 1 to infinity. If your formula does start at 0, just subtract off the the "0" term which is 1.

For (II) you just have the finite sum \(\displaystyle \sum_{x=1}^6 cx= c+ 2c+ 3c+ 4c+ 5c+ 6c= 1\).

So in order for it to be valid pdf, the sum must be 1?

 

[mmm4444bot]

I deleted your thread on the Statistics board, as it appeared to be an edited duplication of this thread.

 

[girodd]

So in order for it to be valid pdf, the sum must be 1?

i
Any function with non-negative values can be a probability "density" function on a specified domain if the sum of all its values over the domain (for a discrete function like these) or the integral over the domain in the case of an integrable function comes out to be 1.