What Is the Variance of the Number of Claims Filed in a Poisson Distribution?

[buzzmath]

Homework Statement

An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed?


2. Homework Equations [/]
P(x)=e^(-lamda)*(lamda^x)/x! Var(x)=lamda

The Attempt at a Solution

I know that P(2)=3P(4) and that P(2)=e^(-lamda)*(lamda^2)/2 and that P(4)=e^(-lamda)*(lamda^4)/4! So I set these equal to find lamda and that is the variance. The solution given to me is P(2)=e^(lamda)*(lamda^2)/2=3*e^(-lamda)*(lamda^4)/4!=3*P(4)*24*lamda^2=6*lamda^4 I don't understand the last two equalities. Any help? where do the 24*lamda^2 comefrom and how do they get 6*lamda^4?
Thanks

 

[buzzmath]

Is this a typo? if so how would I go about solving the problem using the method I've started? or any other way?
Thanks

 

[Mark44]

From the given information,
P(2) = \frac{e^{-\lambda}\lambda^2}{2!} = 3P(4) = 3\frac{e^{-\lambda}\lambda^4}{4!}
From this, we get
\frac{e^{-\lambda}\lambda^2}{2} = 3\frac{e^{-\lambda}\lambda^4}{24}
so
\frac{e^{-\lambda}\lambda^2}{2} = \frac{e^{-\lambda}\lambda^4}{8}
or
4e^{-\lambda}\lambda^2} = e^{-\lambda}\lambda^4}
That works out to 4\lambda^2 = \lambda^4
Can you take it from there?

BTW, the name of this Greek letter is lambda.