Upper bound for $P(\sum_{1}^{20} X_i>15)$ with $X_i\thicksim Poisson(1)$

Question

I have this question

Let $X_i\thicksim Poisson(1)$ identical independant random variables. Obtain an upper bound for $P(\sum_1^{20}X_i>15)$ without using the Central Limit Theorem.

Using it I saw the upper bound should be $0.025$, but I don't know another method to obtain an upper bound. Could anyone give me a hint to obtain it?

Answer 1

Hint:
Sum of independent Poisson variates is itself a Poisson variate, so you need to estimate the cumulative distribution function of a Poisson r.v. with mean $20$. Which is essentially bounding the partial sum of the exponential series... you should get $\approx 0.84$