Let $X_1, \dots, X_n$ be i.i.d. random variables with expectation $a$ and variance $\sigma^2$, taking only positive values. Let $m < n$. Find the expectiation of $\displaystyle\frac{X_1 + \dots + X_m}{X_1 + \dots + X_n}$.
My attemps to solve this probles are rather straightforward. Denote $X = X...
I mean, this is little different from what you usually encounter in calculus class, such as $$ \int_{0}^{1} \int_{0}^{1} x \, dx dy = \int_{0}^{1} \int_{0}^{1} y \, dx dy $$
If we have to be nitpicking, it is a change of variables $(x_1, x_2, \cdots, x_{i-1}, x_i, x_{i+1}, \cdots, x_n) \to (x_i, x_2, \cdots, x_{i-1}, x_1, x_{i+1}, \cdots, x_n)$ and a small touch from Fubini's theorem
As I saw there was no probability and statistics chat room so I thought it would be good creating it,let's see where this goes!
Thanks!
room topic changed to Probability and Statistics: Any discussion on Probability and Statistics.For rendering LaTeX math.ucla.edu/~robjohn/math/mathjax.html [,] [,] [probability-theory] [random-variables] [statistics]
room topic changed to Probability and Statistics: Any discussion on Probability and Statistics.For rendering LaTeX math.ucla.edu/~robjohn/math/mathjax.html [probability-theory] [random-variables] [statistics]
Prove the following statement:
Let $X_{1},X_{2}, \dots ,X_{n}$ be a set of exchangeable random variables. Then, $$E\left(\frac{X_{1}+X_{2}+X_{3}+\dots +X_{k}}{X_{1}+X_{2}+X_{3}+\dots +X_{n}}\right) = \frac{k}{n} , \qquad 1 \leq k \leq n. $$
I tried writing $\frac{X_{1}+X_{2}+X_{3}+...+X_{k...
Prove the following statement:
Let $X_{1},X_{2}, \dots ,X_{n}$ be a set of exchangeable random variables. Then, $$E\left(\frac{X_{1}+X_{2}+X_{3}+\dots +X_{k}}{X_{1}+X_{2}+X_{3}+\dots +X_{n}}\right) = \frac{k}{n} , \qquad 1 \leq k \leq n. $$
I tried writing $\frac{X_{1}+X_{2}+X_{3}+...+X_{k...
Probability and Statistics
Probability and Statistics