Kolmogorov proved, that, as one considers independent (not necessary equally distributed) Random Variables: $\{X_n\}_{n\ge0}\subseteq \mathcal L^2$ With $\mathrm{Var} (X_n)=\sigma^2_n$ and without loss of generality $E[X_n]=0$. If $\sum_{n=0}^\infty \frac{\sigma^2_n}{n^2} \lt \infty$ then SLLN holds, that is: $$\frac1n\sum_{k=0}^n X_k \rightarrow 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\text{a.s.}$$ Something different: Also it is known, that if $\sup_{n\ge0}\sigma^2_n =: v \lt \infty$ pairwise uncorrelated […]

I am designing a virtual card game and I defined an evolution of probabilities, but I don’t have the knowledge on this matter to find out how they will evolve. I hope you help me here, with bibliography and teaching me some things. This is the system: I have a discrete distribution of probability of […]

Is there anyway to solve $\displaystyle \int t \frac{\sin \left(\frac{t}{2} \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}\right) }{ \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}} \operatorname{d}t \tag8$ either by analytical method or from geometrical method with out using numerical methods? means looking for a closed form with out infinite series expansion in the result NB: Main issue is the lack of […]

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of $L^2$ random variables with expected value $m$ for all $n$. Let $S_n=\sum_{i=1}^n X_i$ and $|\mathrm{Cov}(X_i,X_j)|\leq\epsilon_{|i-j|}$ for finite, non-negative constants $\epsilon_k$. Show that: (1) If $\lim_{n\to\infty} \epsilon_n=0$ then $S_n/n\to m$ in $L^2$ and probability (2) If $\sum_{k=1}^\infty \epsilon_k<\infty$, then $\mathrm{Var}(S_n/n)$ is of order $O(1/n)$ and $S_n/n\to m$ almost surely (1) […]

There are similar posts to this one on stackexchange but none of those seem to actually answer my questions. So consider the CLT in the most common form. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables with $X_1 \in L^2(P)$ and $\mathbb{E}[X_i]= \mu$ and $\mathbb{V}ar[X_i] = \sigma^2>0$. Denote with $\widehat{X}:= \frac{(X_1+\dots+X_n)}{n}$. Then […]

In the coupon collector‘s problem, let $T_n$ denote the time of completion for a collection of $n$ coupons. At time $T_n$, each coupon $k$ has been collected $C_k^{n}\geqslant 1$ times. Consider how often the most frequently chosen coupon, was chosen, that is, the random variable $$C^*_n=\max_{1\leqslant k\leqslant n}C_k^{n}. $$ Can one compute $E(C^*_n)$? What is […]

Ref :Introduction to Mathematical Statistics-Prentice Hall (1994) by Robert V. Hogg, Allen Craig. Now , in the above problem it has been shown that a sequence converges to a random variable X in distribution but the sequence of PMF doesn’t converge to the PMF of X. but we know that “a sequence {Xn} with PDF/PMF […]

I recently asked a question here that was the following: If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = P(Z_i=+1) = \frac 12,$ then we have by the Central Limit Theorem that $\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\stackrel{d}{\to} \mathcal{N}(0,1),$ so that for any continuous bounded function $f,$ we have $\mathbb{E}f\left(\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\right)\to\mathbb{E}f(W)$ where $W\sim\mathcal{N}(0,1).$ Now, $|\cdot|$ is not a bounded function, so […]

Let $(\Omega,\mathcal F,P)$ be a finite measure space. Let $X_n:\Omega \rightarrow \mathbb R$ be a sequence of iid r.v’s I need to prove that if: $ n^{-1}\sum _{k=1}^{n} {X_k} $ converges almost surely to $Y$ then all $X_k$ have expectation. If I understand correctly then $X_k$ has expectations means $X_k$ is in $\mathcal L^1(\Omega)$. And […]

Assume A,B,C,D, are pairwise independent events. Decide if $(A \cap B)$ and $(B \cap D)$ are independent events? Then repeat this assuming the four events are mutually independent. Well, what I’m thinking is $P(A \cap B) = P(A) \cdot P(B) \\P(A \cap C) = P(A) \cdot P(C) \\P(A \cap D) = P(A) \cdot P(D) \\P(B […]

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