Actuarial Outpost SOA # 147 E(X)= lambda? 1/lambda^-1?
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#1
08-12-2018, 04:20 PM
 yhanc17 SOA Join Date: Jun 2018 Posts: 2
SOA # 147 E(X)= lambda? 1/lambda^-1?

I looked up SOA solution and the Ex=lambda and Variance=lambda^2.

But Wikipedia and the book I bought: Probability for Dummies says Ex of the exponential distribution is lambda^-1 and variance is 1/lambda^-2.

Can someone please tell me which one is right? I don't understand why the one says one thing and the other says the other.

And I searched the forum and everyone is assigning random variable Y to the amount of claim after deductible.

Is additional random variable needed in this question? I just started studying probability and I am not that far in understanding the entire probability theory yet. I am sorry about asking what seems to be a basic for some people.

Thank you.

Last edited by yhanc17; 08-13-2018 at 08:24 AM..
#2
08-12-2018, 04:33 PM
 Michael Mastroianni Member SOA Join Date: Jan 2018 Posts: 35

There are two common ways of parameterizing an exponential:

(1) If you write the pdf in the form $\lambda e^{-\lambda x}$ then the expected value is $\frac{1}{\lambda}$.

(2) If you write the pdf in the form $\frac{1}{\theta}e^{-x/\theta}$ then the expected value is just the parameter $\theta$ which is nice.

It might seem a bit weird to use the first form and call the parameter of the exponential $\lambda$, but the reason is because of its connection to the Poisson with parameter $\lambda$ in the context of a poisson process.

To be precise: If the number of events in any time interval of length $t$ is Poisson with parameter $\lambda t$ and the number of events in disjoint intervals are independent, then the time between any two events is exponential with parameter $\lambda$ as in (1) above. That's the connection.

EDIT: You have misquoted wiki above. The variance of an exponential is always the square of its mean. In (1) they are $\frac{1}{\lambda}$ and $\frac{1}{\lambda^2}$. In (2) they are $\theta$ and $\theta^2$. I hope that helps.
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Last edited by Michael Mastroianni; 08-12-2018 at 05:37 PM..
#3
08-13-2018, 08:26 AM
 yhanc17 SOA Join Date: Jun 2018 Posts: 2

Quote:
 Originally Posted by Michael Mastroianni There are two common ways of parameterizing an exponential: EDIT: You have misquoted wiki above. The variance of an exponential is always the square of its mean. In (1) they are $\frac{1}{\lambda}$ and $\frac{1}{\lambda^2}$. In (2) they are $\theta$ and $\theta^2$. I hope that helps.
Oops, I am sorry. I fixed the 1/lambda^-1 part. And thank you for the clarification. I got it now.