The problem:
If 48 random numbers are selected independently from the interval(0,1),
what is the approximate probability that the sum of these numbers is at least
21?
The question:
Actually what I want to ask here is that how to find the EXi and the variance, I have the solution but they don't give the explanation so I wonder how to find it. The EXi is 1/2 and the variance is 1/12 but I don't know where it comes from. I hope somebody could help me, any bits of helps would be very appreciated. Thank you for your kindness guys and have a nice day.
The quoted problem is sloppily stated: it does not specify the sampling distribution of the summands.
But it is probably uniform on $[0,1]$. That is, the probability density function is $f(x)=1$ if $x\in[0,1]$ and $0$ otherwise, so $P(a<X<b)=b-a$ for all $0\le a\le b\le0$. . Then the stated mean and variance follow from these calculus facts: $$\int_0^1 x\,dx = 1/2$$ and $$\int_0^1 x^2\,dx -\left(\int_0^1 x\,dx\right)^2 = 1/3-1/4 = 1/12.$$