$X\sim U(0,1)$. Divide the interval [0,1] into k equal subintervals. Then $X_1$=the number of observtions on the first interval.
Define the new variable $Y_1=X_1/n$, where n is the number of observations, and $Y_1$ then is the proportion of the observation on the first interval.
What is the expected value, variance and standard deviation for $Y_1$?
From the definition of the expected value we have $E(Y)=\int\limits_\infty^\infty\mathrm{y}f(y)\mathrm{d}y$, if I could get help with this I guess i can do the variance and standard deviation.
But how do i find $f(y$)?
Best Answer
I take it that you're drawing $n$ independent samples from $X\sim U(0,1)$. Then $X_1$ has a binomial distribution with parameters $n$ and $p=1/k$. The mean and variance of a binomial distribution are known, and $E[Y_1]=E[X_1]/n$ and $\operatorname{Var}(Y_1)=\operatorname{Var}(X_1)/n^2$.