I should solve the following:
Let $X_i,...,X_n$ be independent random variables such that for any i, j, i$\neq$j: $P(X_i=X_j) = 0$. Then, $X_i$ is an upper record iff $X_i > max$ {$X_1,X_2,...,X_{i-1}$}.
So, $R_n = Y_1 + Y_2 + ... Y_n$ denotes the number of upper records in a sequence, here Y_i = (i) 1 if $X_i$ is a upper record and (ii) 0 otherwise (Indicator-Function).
Tasks:
- Compute $E(R_n)$ and the limit of $E(R_n)$. Interpret the last limit result. How looks the probability space for this setup?
- What is the variance $V(R_n)$ of the total number of upper records $R_n$?
EDIT
@1:
$P(X_i = record)$ = ?
By permutations: In a sequence of n values $X_1,...,X_n$ there are n! permutations (= ways, to mix them). If $X_n$ = max, then there are (n-1)! permutations for the remaining $X_1,...,X_{n-1}$.
Probability space ($\Omega, \mathcal F, \mathbb P$) for $R_n$:
$\bullet$ $\Omega = \{1,2,3,4,5,6,...\} = \mathbb N_{>0}$ ($0$ not included because $X_1$ is always a record and therefore $Y_1$ = $1$)
$\bullet$ $\mathcal F = P(\mathbb N_{>0}) $ (Powerset with $2^{N_{>0}} possibilities)$
$\bullet$ $\mu=\Sigma^{n}_{i=1} \frac{1}{i}$ ??
$E(R_n) = E(\Sigma^{n}_{i = 1} Y_i) = E(\Sigma^{n}_{i=1} \mathcal I_{x_i = record}) = \Sigma^{n}_{i = 1}E(\mathcal I_{x_i = record}) = \Sigma^{n}_{i = 1} \frac{1}{i} $
$\lim_{n\to \infty} \Sigma^{n}_{i = 1}$ $1 \over i$ = $\infty$ (harmonic series)
Interpretation of limes: The expectation exists, but is infinite, so one could argue, that's because we do not know how many records we will have in the future. Further we have learnt, that an absolute convergence is enough for an expectation, meaning, the expectation can still be infinite, but existing.
@2:
$Var(R_n) = Var(\Sigma^{n}_{i=1} Y_i) = Var(\Sigma^{n}_{i=1} \mathcal I_{x_i = record}) = \Sigma^{n}_{i=1} Var(\mathcal I_{x_i = record}) = \Sigma^{n}_{i=1} (\frac{1}{n} * (1 - \frac{1}{n})) = \Sigma^{n}_{i=1}(\frac{1}{n} - \frac{1}{n^2})$