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A probability provides a quantitative description of the likely occurrence of a particular event.

7
votes
8/21 is the proportion of heads in the result. Instead of calculating the probability of 8 heads, you can calculate the probability that the proportion of heads in 21 coin flips will be 8/21. They both turn out 0.097 (assuming your calculation is correct) …
answered Jul 31 '14 by BCLC
2
votes
$$ Suppose $$E[X-E[X])^2] = 0$$ Then $$(X-E[X])^2 = 0 \ \text{a.s.}$$ The last step I believe involves continuity of probability...or what you did (You are right). Theres's also Chebyshev's …
answered Dec 4 '15 by BCLC
4
votes
$$P(X^2 > a) = P(X > \sqrt a \cup X < -\sqrt a) = P(X > \sqrt a) + P(X < -\sqrt a)$$ The hypothesis can be weakened to $a \ge 0$
answered Dec 23 '15 by BCLC
0
votes
What is meant by separate? Disjoint? Do you mean $A \cap B = \emptyset$? If so, then your claim is equivalent to: If $A \cap B = \emptyset$, then $P(A \cap B) \ne P(A) P(B)$ To try to prove thi …
answered Sep 18 '15 by BCLC
4
votes
1answer
Consider the following random variables in $(\Omega, \mathfrak{F}, P)$. a $X_1,X_2, X_3,...$ where $\forall n \in \mathbb{N}, \mu_n = E(X_n), \sigma_n^2 = Var(X_n) < \infty$ b $N$, a discrete RV who …
asked Oct 4 '14 by BCLC
1
vote
2answers
From Itō isometry Wikipedia page: Can we switch the expectation and integral in the RHS by Fubini's theorem? I'm actually not quite sure how Fubini's theorem is used outside basic Calculus. The f …
asked Sep 13 '15 by BCLC
1
vote
1answer
From Williams' Probability w/ Martingales: Are we allowed to switch derivative and integral as follows: $$\frac{\partial}{\partial \lambda} \int_{0}^{\infty} e^{-\lambda x} f(x) = \int_{0 …
asked Jul 9 '16 by BCLC
0
votes
1answer
Probability with Martingales: What I tried: I think the hint is equivalent to $$\{\omega | X_n \rightarrow X\} = \bigcap_{k=1}^{\infty} \{\omega | \liminf_n [|X_n - X| \le \frac{1}{k}]\}$$ …
asked Jul 8 '16 by BCLC
8
votes
1answer
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture: Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ or $1 …
asked Dec 23 '15 by BCLC
1
vote
1answer
almost surely. Is it really only almost surely? What is an example of a filtered probability space and event $A$ s.t. $\exists t \in \mathbb N$ s.t. $P(P(A|\mathscr F_t) = 0) = 1$ or $P(P(A|\mathscr …
asked Jan 24 '16 by BCLC
-1
votes
1 No 2 Yes Definition: A family $\mathscr F$ of events is independent if, for every finite number of distinct events $A_1$, $A_2$, $\ldots$, $A_n$ in $\mathscr F$, $$P\left(\bigcap_{i=1}^nA_i\right …
answered Dec 23 '15 by BCLC
2
votes
0answers
From Probability with Martingales: I chose $\mathscr F_n = \sigma(B_1, B_2, ..., B_n)$. My argument assumes that $E[M_n | \mathscr F_{n-1}] = E[M_n | B_{n-1}]$. I was able to show that $M_{n … not, might it help if I define $B_n = X_1 + ... + X_n$ where $X_i$'s are iid Bernoulli conditioned on some probability p (or $\Theta$?) and then use a similar argument to the one here? (I'm guessing …
asked Dec 27 '15 by BCLC
0
votes
Important inequalities Williams - Probability with Martingales Deduced similarly: (iii) If $\liminf x_n > z$, then $ \ \ \ \ \ \ \ (x_n > z)$ eventually (that is, for infinitely many n …
answered Jul 20 '16 by BCLC
-2
votes
1answer
This is supposed to be related to the 2nd Borel-Cantelli Lemma (my justification for the independence tag). In Williams' Probability with Martingales, 2BCL is proven and then the following is given …
asked Sep 12 '15 by BCLC
1
vote
zhoraster's answer (link added by me): Hint for the case $S<1$: Prove this by induction for a finite number of probabilities using that $(1-a)(1-b)\ge 1-a -b$ and then proceed to the limit. …
answered Sep 14 '15 by BCLC

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