# Consider a set of 20 independent and identically distributed uniform random variables in the interval (0,1). What is the expected value and variance of the sum, S, of these random variables? Select one: a.E(S)=10, Var(S)=1.66667 b.E(S)=200, Var(S)=16.6667 c.E(S)=100, Var(S)=200 a.E(S)=10, Var(S)=12

Consider a set of 20 independent and identically distributed uniform random variables in the interval (0,1). What is the expected value and variance of the sum, S, of these random variables?
Select one:
a.$E\left(S\right)=10$, $Var\left(S\right)=1.66667$
b.$E\left(S\right)=200$, $Var\left(S\right)=16.6667$
c.$E\left(S\right)=100$, $Var\left(S\right)=200$
a.$E\left(S\right)=10$, $Var\left(S\right)=12$
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Step 1
Uniform distribution:
A continious random variable X is said to follow Uniform distribution if the probability density function of x is,
$f\left(x\right)=\frac{1}{b-a}$, where a is lower limit and b is the upper limit of x.
The mean of the Uniform distribution is $\mu =\frac{a+b}{2}$.
The variance of the Uniform distribution is ${\sigma }^{2}=\frac{{\left(b-a\right)}^{2}}{12}$.
Step 2
In this case, the iid random variables ${X}_{1},{X}_{2},\dots ,{X}_{20}$ follows uniform distribution over the interval (0, 1).
Consider a new random variable $S={X}_{1}+{X}_{2}+\dots +{X}_{20}$.
Here, it is needed to find the mean and variance of S.
Hence, the mean of Y is,
$E\left(S\right)=E\left({X}_{1}\right)+E\left({X}_{2}\right)+\dots +E\left({X}_{20}\right)$

$=20\left(\frac{0+1}{2}\right)=10$
Hence, the variance of S is,
$V\left(S\right)=V\left({X}_{1}\right)+V\left({X}_{2}\right)+\dots +V\left({X}_{20}\right)$

$=20\left(\frac{{\left(1-0\right)}^{2}}{12}\right)=1.66667$
Thus, the correct option is a.

Jeffrey Jordon