Can someone explain why and how the following is true?
Let us say we have $n$ IID Chi-square random variables $X_i \sim \chi^2(2) \ ; i=1,\cdots,n$, and $Z := \sum_i X_i$, then $$\Pr\left\{ Z < a\right\} = \Pr\left\{ \sum_{i=1}^n X_i < a\right\} \geq \left(\Pr\left\{X_1 < a\right\}\right)^ n .$$
From the discussion below: The above inequality is not true. It should be $$\Pr\left\{ \sum_{i=1}^n X_i < a\right\} \leq \left(\Pr\left\{X_1 < a\right\}\right)^ n .$$
Since from definition $X_i$ are nonnegative, $\sum_i X_i <a$ implies each $X_i<a$ and then you can use the fact that they are independent to provw the inequality.
That means $\{\sum_i X_i<a \}\subset \cap_i\{X_i<a\}$ and we can take their measures and use independence of these events $\{X_i<a\}$