1
vote
1answer
9 views

Prove that the discrete time martingale can be represented by $E (Y_{n +1} \mid F_n) = 0$ if $Y_{n +1} = X_{n +1}-X_n$, for $n = 0,1, \ldots $

Prove that the discrete time martingale can be represented by $E (Y_{n +1} \mid F_n) = 0$ if $Y_{n +1} = X_{n +1}-X_n$, for $n = 0,1, \ldots $ I want to use the sequence $(y_n)$ called "martingale ...
0
votes
1answer
35 views

Calculate the Probability in competition

The committee RAM competition knows from experience that the probability of successfully Contest is 0.95 for the student who has grade "very good" in BAC test , 0.5 one who has 'Good' in ...
0
votes
0answers
38 views

Probability of getting head in coin flip [duplicate]

Suppose a football match is going to be started. The referee should flip a coin to give the ball to one of the football teams. In front of the two captains the referee flips the coin two times and ...
0
votes
1answer
36 views

Show that $X_n \stackrel{d}{\longrightarrow} 0$ iff $\{\varphi_n(t)\}$ converges to 1 in some neighbourhood of $t=0$.

$X_n$ is a sequence of random variables, and $\{\varphi_n(t)\}$ is the corresponding sequence of characteristic functions. Show that $X_n \stackrel{d}{\longrightarrow} 0$ iff $\{\varphi_n(t)\}$ ...
1
vote
0answers
21 views

Infinite fourth moment and maximum entropy

Alright, I expect this is a silly question, but I don't actually know, so. Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean ...
0
votes
1answer
63 views

Probability two people born on 1 April

Find the smallest number of people you need to choose at random so that the probability that at least two of them were both born on April 1 exceeds 1/2. My answer: Total of n people chosen at random ...
1
vote
1answer
28 views

Probability density function of two uniformly distributed stochastic variables

I'm currently stuck on an exercise involving two independent stochastic variables X and Y. Both X and Y ~ U(0,1) (uniform distribution) The goal of the exercise is to calculate the probability ...
0
votes
1answer
23 views

Distributions of local times of a single excursion of 1D random walk

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
0
votes
1answer
84 views

Expected number of coin tosses until a run of $k$ successive heads occurs

Suppose each coin toss is independent, what is the expected number of coin tosses until a run of "k" successive heads occur? Tried finding a recursive expression to solve the problem but got ...
6
votes
3answers
135 views

Birthday “Paradox” - another, different, version!

Background Many people are familiar with the so-called Birthday "Paradox" that, in a room of $23$ people, there is a better than $50/50$ chance that two of them will share the same birthday. In its ...
0
votes
1answer
38 views

Help proving $Pr(\mathcal{X})= \phi_1(X,Z)\phi_2(Y,Z)$ if $ P \models (X \perp Y | Z)$ and $\mathcal{X}=X \cup Y \cup Z$

I was trying to prove the following: if $X,Y,Z$ were three disjoint subsets of variables such that $\mathcal{X}=X \cup Y \cup Z$, Prove that $ P \models (X \perp Y \mid Z)$ if and only if we can ...
0
votes
2answers
22 views

Laplace transform of noncentral chi-square distribution

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...
1
vote
1answer
41 views

Is there a set of 69 length-6-sets out of 46 numbers [1..46] so that those length-6-sets “cover” all possible 1035 length-2-sets of 46 numbers?

1.) For this question, we have 46 numbers (balls, cards, whatever): {1,2,3,4 .... 45,46} ======================= 2.) Each length-6-set of 46 numbers ( e.g. {1,2,3,4,5,6} or {1,13,16,17,32,46 } ...
0
votes
0answers
9 views

Generate quadrature points from a distribution

Is there any method to generate quadrature points from any arbitrary probability distribution, $p_{X}\left(x\right)$? We already know about Gauss Hermite rule for Normal distribution, Gauss-Laguerre ...
0
votes
2answers
33 views

Total boundness of Lipschitz densities

In the article Almost Sure Testability of Classes of Densities by Devroye and Lugosi in 1999. They claim in Example 10 (page 9) that Lipschitz densities on [0,1] with Lipschitz constant bounded by ...
0
votes
1answer
15 views

How to solve disjoint probability problem?

Suppose the two events “high” and “low” make a disjoint partition of a sample space and “favourable” is any event. If P(high) = 0.3, P(low) = 0.7, P(favourable| high) = 0.9 and P(unfavorable| low) = ...
1
vote
2answers
33 views

(What is the formula to find) What is the probability that the sum of the numbers on the tickets chosen is at least 7?

Senario: Box A contains four equal-sized tickets, numbered 1, 2, 3 ,4 Box B contains three tickets of the same size, numbered 4, 5, 6 An experiment consists of selecting one ticket from the box A ...
1
vote
0answers
33 views

Properties of the essential supremum

In the article about testability of densities (Devroye and Lugosi 1999, page 4) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ). They use some properties of the ...
2
votes
1answer
60 views

Example of non continuous random variable with continuous CDF

Can someone provide an example of $X$ being a non-continuous random variable with continuous cumulative distribution function? For instance: $X$ is discrete if it takes (at most) a countable number ...
0
votes
0answers
40 views

What did I do wrong when using Jacobian transformation

A device containing two key components fails when, and only when, both components fail. The lifetimes, $T_1$ and $T_2$, of these components are independent with common density function $f (t) = ...
0
votes
2answers
32 views

probability of a flipped coin

A fair coin is flipped three times. Let $A$ be the event that a head occurs in the first flip and $B$ be the event that exactly one head occurs. a) Find $p(A/B)$ b) Are $A$ and $B$ independent? ...
-1
votes
2answers
34 views

If pages in a book have an iid Poisson number of errors, in 10 pages what is the probability that exactly 3 pages have exactly 1 error?

Suppose the number of spelling error on any given page in particular book can be modeled by a Poisson distribution with $\lambda=2$, and assume that the number of errors on different pages is ...
0
votes
0answers
38 views

Kelly criterion for 3 outcomes

I have been exploring the Kelly criterion for optimizing the bet size for a two outcome bet situation. I'm having trouble applying this to a three outcome bet. I may refer to this excellent thread: ...
2
votes
2answers
85 views

Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
2
votes
1answer
40 views

A question about Malliavin calculus

An application of Malliavin calculus is to calculate the sensitivity of financial Greeks. However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to ...
0
votes
1answer
22 views

Polynomial joint pdf $f(x,y)$ such that of $f(x) \neq f(y)$

How can I build a polynomial joint pdf $f(x,y)$ for $x \in [x_1, x_2]$ and $y \in [y_1, y_2]$ such that of $f(x) \neq f(y)$ or equivalently, $x$ and $y$ are depended on each other?
5
votes
4answers
137 views

What is the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$ and some of the properties of $E[X \mid Y]$?

I was trying to understand both intuitively and rigorously what the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$. Let me tell you first the things that do make sense to me. $E[X\mid Y=y]$ makes ...
0
votes
1answer
36 views

Show that $Pr[X \gg Y]\approx 1$

Can one show (and how) that $$Pr[X \gg Y]\approx 1$$ for $$X:=\sum_{i=1}^k Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ and $$Y:=\sum_{i=k+1}^{\infty} ...
0
votes
0answers
63 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
1
vote
2answers
94 views

What is the probability of going bankrupt in roulette?

Imagine that the bank has the money $M_1$ and the player has the money $M_2$. The rules are the following: You win with a chance of $\frac{17}{36}$ and lose with $\frac{19}{36}$ each round. Now you ...
1
vote
0answers
30 views

On an existence of a real-valued measurable function on a non-atomic probability measure space [duplicate]

Suppose that $(X,E,μ)$ is a non-atomic probability measure space. Let $\xi :X \to \mathbb{R}$ be a random variable. A Borel measure $\mu_{\xi}$ in $\mathbb{R}$ defined by ...
1
vote
0answers
38 views

Joint probability of 3 random variables when their pairwise difference is given

Consider 3 discrete random variables $X_1,X_2,X_3$ defined over $\{0..T\}$, which are identically and uniformly distributed.They are correlated in the sense that their pairwise difference has a unique ...
2
votes
2answers
42 views

Probability that exactly 2 of 3 objects are in 1 of 3 baskets with sizes 5, 8, 2

I want to calculate the probability that some mutation occurs on a certain DNA section by a given number mutations. I rephrased it into this problem: Three (identical) persons enter a train (section ...
1
vote
2answers
76 views

Partial sum of binomial

I 'm trying to figure out a closed form solution for the following summation: $\sum_{j=0}^{\omega} j{n \choose j}p^{j}(1-p)^{n-j}$ where $\omega < n$ Is there any closed form solution?
0
votes
0answers
42 views

Special case of Kullback-Leibler additivity

I have three random variables $X,Y,Z$. If $(X,Z)$ are an independent pair and $(Y,Z)$ are an independent pair, then the additive property of the Kullback-Leibler divergence says $K(X,Z|Y,Z) = K(X|Y) ...
2
votes
1answer
40 views

Expected Payment under limited policy

The unlimited severity distribution for claim amounts under an auto liability insurance policy is given by the cumulative distribution: $$ F(x) = 1 - 0.8e^{-0.02x}-0.2e^{-0.001x} , x \geq 0$$ ...
0
votes
1answer
35 views

Question about exp. distribution

We know that $X\sim \exp(1),Y\sim \exp(2)$ and they are independent. What is $P(Y>X)$? exp=Exponential... Thank you!
0
votes
1answer
26 views

Need help with P(D) in a Bayesian model

So I've been reading about Bayesian models so I tried I'd have a toy example I could play with. Consider the following: You are at a bus stop and you observe the bus arriving at various times $t_1, ...
1
vote
2answers
29 views

What is Cumulative Binomial probabilities?

I am new to this so don't know if I am asking the right question as I just read about its usage but didn't know what exactly a Cumulative Binomial probability is. So my question is, What is ...
0
votes
2answers
284 views

Two groups A and B are playing a game…

Two groups A and B are playing a game. The first group that wins 3 times is the winner. The probability that group A will win at on game is $\frac12$ and the same thing for group B. $X$ = The number ...
0
votes
0answers
42 views

Is the following probability distribution stationary/constant

For a conservative system, we know that angular momentum, $l$, and total energy, $E$, are constant, i.e. $\dot{l}=\frac{dl}{dt} = 0$ and $\dot{E}=\frac{dE}{dt} = 0$, where $t$ indicates time. Let ...
1
vote
1answer
39 views

Find Limiting Distribution of $|X_n|$

Let $Z_1,Z_2,...,Z_n,...$ be a sequence of independent standard normal random variables. Let $X_n=\sum^n_{k=1}\frac{Z_k}{\sqrt{k}}$. Does the limiting distribution of $|X_n|$ exists? If yes, find it; ...
-1
votes
2answers
31 views

We are making a Bernoulli experiment… [on hold]

We are making series of independent Bernoulli experiment with $\frac13$ chance to success. What is the probability that we got success at the first experiment, if we know that we get two successes at ...
0
votes
3answers
278 views

What is the probability that A will win…

Two players are rolling two dices, if they get 6 Player A wins, if they get 7, player B wins, else they rolling the two dices again... What is the probability that A will win? I'd like to get any ...
0
votes
1answer
34 views

Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
3
votes
1answer
56 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
1
vote
2answers
32 views

Need help understanding the difference between a.s. convergence and convergence in probability.

I have problem understanding the difference when I look at the alternative definition of a.s. convergence. I know how it is defined originally, but it is the alternative definition which makes it ...
2
votes
1answer
108 views

Deducing an optimal gambling strategy (using martingales).

Apologies in advance for the length, I tried being precise. Suppose a game where in each turn you can gamble a certain amount of money on the result of a fair coin toss. If the coin comes out tails ...
-1
votes
1answer
24 views

Asymoptotic distribution of identically distributed random variables [closed]

$Y_1, Y_2, ..., Y_N$ are independent and identically distributed random variables with the distribution function $F := F_{Y_1}$ and $F'_n(y) = \frac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{\{Y_i \leq x\}}$ as ...
2
votes
0answers
56 views

How is conditional probability being used here?

Because of conditional probability: $P(A\mid B)=P(A,B)/P(B)$, $$P(C(t)\in dt\mid x(T^+_{i-1}),x(T^-_{i}))=\dfrac{P(C(t)\in dt,x(T^-_{i})\in dx\mid x(T^+_{i-1}))}{P(x(T^-_{i})\in dx\mid ...