Finding the PDF of Y

Question

I have this equation and from this I need to obtain the PDF of Y.

I have the solution but the steps aren't clear.

The answer I have is

But I have been given no steps - any help would be greatly appreciated.

Answer 1

There are too many typographical errors in the provided solution, and now you still have to evaluate two summations in the solution. Less painful would be to write out the summations from the get-go: $$ 1- F(t) = \sum_{u=0}^2 e^{-\lambda t}\frac{(\lambda t)^u}{u!}=e^{-\lambda t}\left(1 + \lambda t +\frac{(\lambda t)^2}{2!}\right)\tag1 $$ Having cleared out the summation notation, now differentiate both sides of (1). Use the product rule on the RHS of (1) to obtain $$ -f(t)=-\lambda e^{-\lambda t}\left(1 + \lambda t +\frac{(\lambda t)^2}{2!}\right)+e^{-\lambda t}\Big(0 + \lambda +\lambda(\lambda t)\Big).\tag2 $$ Multiply (2) through by $-1$, and simplify: $$ f(t)=\lambda e^{-\lambda t}\left(1 +\lambda t+ \frac{(\lambda t)^2}{2!}-1 -\lambda t\right)=\lambda e^{-\lambda t}\frac{(\lambda t)^2}{2!}. $$