Exponential Probability of Sum

Question

Given :

• $X_1, X_2, X_3, \ldots,X_{10}\ \sim\ \,\mathrm{e}^{\lambda}$
• $X_1, X_2, X_3,\ldots,X_{10}\,\,\, \mbox{are independent variables}$
• $\lambda > 0.5$

Calculate the following:

$$ P\left(\sum_{i = 1}^{10}X_{i} \leq {15 \over \lambda}\right)$$

Answer 1

If, $X_i$, $i=1,2,\cdots ,n$, be i.i.d exponential random variables with $\lambda$ parameter $Y=\sum\limits_{i=1}^{n}{{{X}_{i}}}$ has gamma distribution with parameters $n$ and $\lambda$, Indeed $${{f}_{Y}}(y)=\lambda {{e}^{-\lambda y}}\frac{{{(\lambda y)}^{n-1}}}{(n-1)!}$$ therefore $$ F_Y\left(\frac{15}{\lambda}\right)=P\left(Y\le \frac{15}{\lambda}\right)=\int_{0}^{\frac{15}{\lambda}}f_Y(y)dy=\frac{\lambda^{10}}{9!}\int_{0}^{\frac{15}{\lambda}}y^{9}e^{-\lambda y} dy$$