# Finding the given probability(Uniform distributions involved).

We are given that numbers are selected at random from the interval $(0,1)$. If $100$ numbers are selected what is the probability that the average of the numbers is less than $0.5$?

We let $X_i$ be the $i$-th number selected, assuming $X_i\sim U(0,1)$, we need to find $$P\left(\dfrac{X_1 + X_2+ \dots +X_{100}}{100} < 0.5\right)$$ i.e $P(X_1 + X_2+ .... +X_{100} < 50)$.

Assuming $X_i$'s as independent, how can we find the given probability? I don't think $\sum_{i=1}^{n} X_i$ will come out to be a uniform distribution too. Can anyone help ?

• If the numbers $X_i$ are independent and uniform on $(0,1)$ then so are the numbers $1-X_i$, hence the desired probability is $\frac12$ by symmetry. – Did Nov 22 '16 at 19:33

Comment: @Did's answer is simple and does not involve any approximation.

However, if you are in a beginning probability course and just covered the Central Limit Theorem, there is a chance that you are intended to use that. Each $X_i$ has $E(X_i) = 1/2$ and $Var(X_i) = 1/12.$

Therefore $S = \sum_{i=1}^{100} X_i$ has $E(S) = 50,$ $Var(S) = 100/12,$ and $SD(S) = 2.886751.$ By the CLT, $S$ is approximately normal. So you could standardize $S$ and use the (symmetrical!) normal distribution to get the answer.

This method has the advantage that it could also be used to find $P(S < a)$, for numbers $a$ other than 50.

A simple simulation (in R statistical software) makes it possible to illustrate that the distribution of $S$ is nearly normal.

m = 10^4;  n = 100;  x = runif(m*n)
DTA = matrix(x, nrow=m) # each row a sample of 100
s = rowSums(DTA) # vector of sums of the m samples
mean(s < 50)
## 0.499  # aprx P(S < 50),  proportion of s-values below 50

hist(s, prob=T, br=20, col="wheat", ylim=c(0,.15),
main="Simulated Distribution of Sums of 100 UNIF(0,1) Observations")
curve(dnorm(x, 50, sqrt(100/12)), lwd=2, col="blue", add=T)
abline(v=50, lwd=2, col="red")