probability and physics application problem voltages

Question

We have $n$ voltages $V_1, V_2,\dots , V_n$ that are received in a condensator or sum, such that $V=\sum V_i$ is the sum of received voltages in that point. Every voltage $V_i$ is a random variable uniformly distributed in the interval $[0, 10].$

1. Calculate expected values and standard deviation of the voltages $V_i.$
2. Calculate probability that the total voltage entrance overpases $105$ volts, for $n = 20, 50, 100.$

I dont need much help with point $2,$ I just need to use the central limit theorem but I need an expected value and standard deviation of point $1.$ I thought in using theorem of big numbers but I am missing something cause I need to get constants for expected and standard deviation, please help.

Answer 1

For $X,Y$: $E[X+Y]=E[X]+E[Y]$. Also, for independent $X,Y$, $\mbox{Var}(X+Y)=\mbox{Var}(X)+\mbox{Var}(Y)$. Can you finish question 1 now?

Answer 2

$Y_i$ has uniform distribution on $[0,10]$ so the mean of $Y_i$ is $\frac 1 {10} \int_0^{10} xdx=5$. Also $EY_i^{2}=\frac 1 {10} \int_0^{10} x^{2}dx=100/3$. Hence the variance of $Y_i$ is $100/3-5^{2}=25/2$.

Answer 3

I am taking the formulas for random variable uniformly distributed. E(X)= (b+a)/2 and Var(X)=$(b-a)^{2}/12$ thanks i will post the rest when i finish the theorem