# Thread: Probability function based on X ~ Exp(B)

1. ## Probability function based on X ~ Exp(B)

I have a problem which im having difficulties to solve. We have a continuous variable X = time measured in hours until event A happens. X is Exp(B)-distributed. Then we have variable Y which is X rounded up, meaning Y can only take on positive whole numbers. I am now asked to write out the probability function for Y. I'm quite certain that the probability distribution would be given by P(Y=k), where k would for example be 1 if X=0,32 hours, which somehow should be based on X. If this is correct how should I go about constructing P(Y=k), k=1,2,3...?

2. ## Re: Probability function based on X ~ Exp(B)

What values of X would give you Y=1?

3. ## Re: Probability function based on X ~ Exp(B)

I'm sorry! I just realized I misunderstood the question I was working with. I'll give it in full:

"Lisa would like to talk with her colleague John who is busy making a recording. Time X (measured in hours) which it takes John to make the recording is Exp(B) distributed. Lisa looks in on John every whole hour to see if he's finished. Time Y which it takes Liza until she can speak to John is then Y = X rounded up to closest whole number. Determine the distribution of Y"

I have not met this type of problem before. I guess I'm looking for some hints on how to approach the problem.

4. ## Re: Probability function based on X ~ Exp(B)

What values of X would give you Y=1?

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lukketotte (09-24-2015)

6. ## Re: Probability function based on X ~ Exp(B)

Y=1 would give 0<X≤1. This indicates that P(Y=k) = P((k-1)< X ≤ k). Could the answer be integrating the probability function of X over k-1 to k?

7. ## Re: Probability function based on X ~ Exp(B)

Yes you get it, and you need to do the integration, unless you can directly use the CDF.

8. ## The Following User Says Thank You to BGM For This Useful Post:

lukketotte (09-24-2015)

9. ## Re: Probability function based on X ~ Exp(B)

Originally Posted by lukketotte
Y=1 would give 0<X≤1. This indicates that P(Y=k) = P((k-1)< X ≤ k). Could the answer be integrating the probability function of X over k-1 to k?
That sounds perfect to me. Like BGM mentions you could also do this by directly using the CDF but the two approaches would be essentially equivalent.

10. ## Re: Probability function based on X ~ Exp(B)

In doing that I'm given e^(-k/B)(e^(1/B)-1). As Y is discrete im guessing this could somehow be interpreted or transformed into a geometric distribution, especially if we name 1/B=λ. Any ideas?

Edit: Y ~ Geo (1-e^λ)

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