Probability question involving Sum of I.I.D Uniform R.v's

Question

I have got a MCQ question which I am unable to solve.

Let $X_1,X_2,X_3,....,X_n \stackrel{i.i.d}{\sim}$ U($-0.5$,$0.5$) and let

T= $X_1+X_2+X_3+....+X_n$.

Suppose $n=100$, Then $P(T^2>25)$ is?

a) $1/2$
b) $1/3$
c) $1/4$
d) $2/3$

How do I solve this problem without the knowledge of Irwin-Hall distribution? I tried to use CLT, but the answer was around 2/25. Please help!

Answer 1

If you want an exact answer, you could check out https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution. But it sounds likes you want an approximate answer using CLT.

CLT is a theorem on the mean of i.i.d. random variables but your $T$ is just the sum of i.i.d. random variables. So lets rewrite

$\frac{1}{n}T = \frac{X_1 + \ \cdots \ + X_n}{n}$

which is approximately distributed $N(\mu,\frac{\sigma^2}{n}) = N(0,\frac{1}{12n})$, where $\mu = 0$ is the mean of $X_i$ and $\sigma = \frac{1}{\sqrt{12}}$ is the standard deviation of $X_i$.

Then, for $n = 100$, $T = 100 (\frac{1}{100}T) \sim 100 N(0,\frac{1}{12n}) = N(0,\frac{100^2}{12n}) = N(0,\frac{100}{12}) = N(0,\frac{25}{3})$. Let $\sigma^* = \sqrt{\frac{25}{3}}$ be the standard deviation of $T$.

Then, $P(T^2>25) = P(T<-5) + P(T>5) = 2P(T<-5) = 2\left(\frac{1}{2} \left[ 1+ {\rm erf}(\frac{-5-\mu}{\sigma \sqrt{2}})\right]\right) \approx 0.0833.$

I'm mot sure where your multiple choice answers are coming from but I believe this is correct. I ran a monte carlo simulation to confirm and I'm also getting around $0.084$. (Code in MATLAB)

% Analytic
mu = 0;
sigma = sqrt(25/3);
prob_a = 1+erf((-5-mu)/(sigma*sqrt(2))); % 0.0833

% Monte Carlo Sim
n=100;
trials = 1e6;

samples = rand(trials,n)-.5; % uniform(-0.5,0.5)

T = sum(samples,2);

prob = sum((T.^2)>25)/trials; % 0.0836