Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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19 views

Every random variable has at least one median

The real number m is called a median of the distribution function F whenever $$ \lim_{y \to m-}F(y)\leq\frac{1}{2}\leq F(m) $$ Show that every random variable has at least one median. I definitely ...
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10 views

Probability that a joyride is overbooked

A bus operator is known that 5 % of the reserved places are not taken up. The entrepreneur sold for a joyride 320 tickets, but in reality there are only 300 seats. 1) What is the probability that ...
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18 views

Covariance of minimum of independent random variables and a constant

I have two random variables $X \sim \min (k, X_1)$ and $Y \sim \min (k, X_2)$ where $X_1$ and $X_2$ are exponential random variables with same rate $\lambda$ independent of each other and $k$ is a ...
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2answers
22 views

Is the following statement true about probabilities and their complements?

I saw the following statement written, but I can't understand why it is true. $$ \dfrac {P(A \text{ and } B)}{P(B)} = \dfrac{P(A)-P(A \text{ and }B^c)}{ 1-P(B^c)} $$ Any help understanding why these ...
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0answers
17 views

Is there any standard way of analysing this integral?

I have a compound Poisson process $(X_t)$, with jump distribution $F$, which assigns mass only to $(0,\infty)$. In my working I have an expression of the following form: $$ \mathbb{E} \int_0^{\tau} ...
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0answers
25 views

Is this function a joint distribution function for two random variables? [duplicate]

Is the following function $F(x,y) = 1-e^{-xy} $ for $x,y \geq 0$ $F(x,y) = 0 $ otherwise a joint distribution function for two random variables $X$ and $Y$? Why?
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12 views

Using Central Limit Theorem in Insurance

An insurer's portfolio contains $2000$ one-year life insurance policies. Half of them are characterized by a payment $b_1= 1$ and a probability of dying within $1$ year of $ q_1 = 1$%. For the other ...
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0answers
16 views

binomial sum bound

Let $X$ be a binomial random variable with parameters $n$ and $p$ that is $X \sim \mathrm{bin}(n,p)$. Does anyone know of a good lower bound (interms of $n$ and $p$) on the probability that $P(X \geq ...
2
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0answers
29 views

a.s. for all $t$ or for all $t$ a.s.?

Assume that we have some equality, $$ X (t) = Y(t). \quad \quad \quad \quad \quad \quad \quad (1) $$ I imagine that if I say "(1) holds a.s. for all $t>0$" it means that $$ P\{X (t) = Y(t) ...
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1answer
11 views

Exercise about conditional probabilities

Exercise: Let $S_1$ and $S_2$ be two disjoint events, and let $\mathcal{G}$ be a sub-$\sigma$-algebra. Show that the following hold with probability $1$. \begin{align*} \mathrm{(a)}& \quad ...
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0answers
16 views

Big O in Stochastic Sense

I understand that if for a real-valued random variable $X$ we have $X = O_p(1)$, then it means that for any $\epsilon>0$, there exists a positive real number $M>0$ such that ...
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1answer
27 views

Convergence of $\operatorname E|X_n|^p$ when $0<p<1$

Let $0<p<1$ and $X_1,\ldots,X_n$ be random variables with finite absolute moments of order $p$. Suppose that the random variables $X_1,\ldots,X_n$ converge in mean of order $p$ to a random ...
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2answers
61 views

Random graph, coin toss probability

We have a coin which comes up heads with probability $p$ at each toss. Let $v_1,v_2,...,v_n$ be $n$ different points on a unit circle. We examine each unordered pair $v_i,v_j$ and toss a coin. If it ...
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0answers
12 views

Compute the kurtosis of a Poisson distribution

I have seen the computation showing that the normal distribution has a kurtosis of 3, but how does one calculate kurtosis if $X$ is Poisson with rate parameter $\lambda$? I know the result should be ...
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0answers
20 views

can anyone solve the question [on hold]

Distinct Object and Distinct Cells (order is considered) Suppose there are n cells and r balls and a cell can have at-most one ball. What are the possible ways to arrange the balls in the cells if ...
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2answers
13 views

Relationship between covariance matrix and its determinant

Let $X=(X_1, X_2,..., X_n)$ be a vector of random variables and define the covariance matrix v to have elements $$ v_{ij} = cov(X_i, X_j) = E(X_i, X_j) - E(X_i)(X_j) $$ Show that the determinant of ...
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0answers
17 views

$F\big( g(t) \big) - F\big( g(t + h) \big) \leq h$ implies that $g$ is right-continuous?

Suppose $F$ is a continuous, strictly increasing distribution function. Also, suppose that $g:[0,1] \longrightarrow [0,1]$ such that for any $t \in [0,1]$, $h > 0$, and $\epsilon > 0$, $$ ...
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1answer
32 views

Complicated Conditional Probability

Three people role a die starting with person 1, then 2, then 3. First to role a 6 is eliminated. Find the probability that B is elimination first. I think this is some summation of probabilities to ...
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3answers
37 views

Simple Probability Question about Combinations

If someone could please point me in the right direction on these. I get lost on how to think about them. In a game there are four holes with values 0, 1, 2, and 4. You are given 6 balls to shoot into ...
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0answers
22 views

Bounding the density of random variable

This is a followup to the question in Bounding the Density of the Maximum of N Random Variables I have a random variable, X, whose cdf is bounded as below: $ \Pr \{X \le x \} \le ...
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1answer
27 views

Positive submartingales

Let $\{X_n\}$, $n>0$ be a positive submartingale with $X_{0} = 0.$ Let $V_n$ be random variables such that $V_n \in\mathcal F_{n−1}$ for all $n \geq 1$. $B > V_1 > V_2 > \dots > 0$ ...
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9 views

Prove that $E_x[T_y]<\infty$ for irreducible positive recurrent Markov chains.

This is an exercise from Durret's "Probability: Theory and Examples". Suppose $p$ is a irreducible and positive recurrent Markov chain. Then $E_x[T_y]<\infty$ for all $x,y$. Since $p$ is positive ...
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0answers
7 views

Volume of randomly changing sphere follows beta distribution

We are given $X,X_1,\ldots,X_N$ independent and identically distributed $k$-dimensional vectors. For a given query point $X_q\in\mathbb{R}^k$ assume without loss of generality that $X_1,\ldots,X_m$ ...
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1answer
27 views

How to show the following about expectations

If $X$ is $\mathcal{M}_1$ measurable on a probability space, $\mathcal{M}_2\subseteq\mathcal{M}_1$ and $Y$ is $\mathcal{M}_2$ measurable and they satisfy $E(XI_A)=E(YI_A)$ for every ...
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2answers
28 views

Conditional probability for a RV with exponential distribution

Let X be a positive random variable such that for all $x,y>0$ we have that $$\mathbb{P}[X >x+y | X>x] = \mathbb{P}[X > y]$$ I need to show that X has exponential distribution, i.e, ...
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1answer
20 views

Are my calculations using Neymann Pearson lemma correct?

I read this post, but I need to use N-P lemma to verify hypothesis doing it really step by step, so please help me. $X_1,X_2,\ldots,X_{30}\sim N(\mu, 1)$, so $\sigma=1$ (I assume that) and $n=30$. ...
4
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3answers
81 views

A recursive formula to approximate $e$. Prove or disprove.

Let the sequence $\{x_n, n=1,2,...\}$ be defined as follows: Let $x_2=x_1=1$ and for $n>2$ let $$x_{n+1}=x_n-\frac{1}{n}x_{n-1}.$$ This sequence, generated by the recursion above, tends to zero ...
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0answers
16 views

If X~N(0,1) and Y~N(0,1) then X+Y~N(0,2) [duplicate]

I am trying to show that If X~N(0,1) and Y~N(0,1) then X+Y~N(0,2), where X and Y are independent random variables. Any help will be appreciated.
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11 views

How can I find P(N(100)>70) where N(t) is a renewal process with interarrival time distributed with f(x) ~ 1/x^4 for x>1?

Since $P(N(100)>70)=P(S_{70}<100)$ but since the arrival time $S_{70}=\sum_1^{70}X_i$ and $f_X(x)=\frac{1}{x^4}$, I will need to find the sum of 70 iid random variable with the above ...
1
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1answer
23 views

Notation of integral.

In some probability book I've come across this notation: $E(X)=\int_{-\infty}^{\infty}x d F(x)$, and it's very confusing, when I see other books defining the same concept as: ...
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1answer
31 views

Find the conditional probability density function $f_{y|x}(y,x)$

Assuming $Z$ random variable with $Z=X+Y$, $Z$ depending on $X$ but is independent of $Y$. We know the value of $X$. Also assume we know the joint pdf $f_{x,z}(x,z)$. Find $f_{y|x}(y,x)$. Can you ...
2
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0answers
19 views

Stochastic control with stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ and smooth functions $u,F: [0, +\infty) \rightarrow \mathbb{R}$, how can we optimise the ...
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1answer
15 views

Condition for strong law of large numbers : Understanding the solution

I am reading the solution of the following task: Here is the solution: From where does the last inequality in the second line comes from? Everything else is totally clear to me.
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1answer
15 views

How is the Binomial coefficient simplified to a falling factorial?

I'm learning how to take the derivative of the binomial coefficient and found a blog post that was quite useful. However I am unclear as to how the first step bellow was simplified to the second step ...
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3answers
51 views

How to comprehend $E(X) = \int_0^\infty {P(X > x)dx} $ and $E(X) = \sum\limits_{n = 1}^\infty {P\{ X \ge n\} }$ for positive variable $X$?

Suppose $X$ is an integrable, positive random variable. Then, if $X$ is arithmetic, we have $E(X) = \sum\limits_{n = 1}^\infty {P\{ X \ge n\} }$ and if $X$ is continuous, we have $E(X) = ...
1
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1answer
35 views

Dual Pairs, topology of weak convergence and weak* topology

I have various questions (e.g. this one) around partly related to the one I am going to ask now. I do have the feeling I finally got what this is all about, but still I am not sure. Thus, here there ...
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1answer
28 views

Measurability of a stopping time in a Markov chain

Suppose you have a finite-state continuous-time inhomogeneous Markov chain with transition rate $Q(t)$. Further, let us suppose that $Q(t)$ is a piecewise continuous function of $t$. Two questions: ...
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0answers
11 views

Looking for books on probability with specific properties

I am looking for book(s) on stochasitcs with a twofold focus. I need the book to derive the basic discrete and continuous distributions, both theoretically and through concrete examples, and generally ...
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1answer
24 views

Box has 10 balls, 6 black, 4 white. Three drawn, color not recorded. What is the probability the fourth ball is white?

In this question we assume the $10$ balls are equally likely to be drawn from the box. What I did was to partition this and say: $P(\text{Fourth Ball is white}) = P(4^{th} \text{ is } W|3 \text{ are ...
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0answers
15 views

When does the 'expected realization' of a stochastic process exist?

Consider a stochastic process $X_t$ (discrete or continuous time). Under what conditions does there exist a process realization $X^{(\mathbf{t})}=\{X_t\}_{t\in[0,\infty)}$ such that: ...
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0answers
28 views

Markov Property Confusion

I feel like I'm being very dense/employing some sort of circular reasoning, but I'm having trouble understanding the Markov Property. According to Durrett (ISBN-10:1461436141), $X_n$ is a Markov chain ...
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1answer
31 views

What are some areas of research/industry involving stochastic processes that aren't finance-related?

I've always enjoyed probability and stochastic processes (took two courses in stochastic models in undergrad, and a PhD level intro to stochastic processes course for my master's). Someday I'd like to ...
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1answer
19 views

Matrix rank and probability limit

If I have a random matrix $M$ (dimension $l\times l$) whose elements are stochastic and depend on $n$ such that for all $n$, the rank of $M$ is less than or equal to $k$, can I infer that $$ ...
4
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1answer
45 views

Convergence types in probability theory : Counterexamples

I know that the following implications are true: $$\text{Almost sure convergence} \Rightarrow \text{ Convergence in probability } \Leftarrow \text{ Convergence in }L^p $$ $$\Downarrow$$ ...
2
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2answers
77 views

Proof of the tower property for conditional expectations

Let $Z$ be a $\mathfrak{F}$-measurable random variable with $\mathbb E(|Z|)<\infty$ and let $\mathfrak{H}\subset \mathfrak{G}\subset \mathfrak{F}$. Show that then $\mathbb E(\mathbb ...
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0answers
18 views

Given the following functions i need to arrange them in increasing order of growth. [on hold]

Given the following functions i need to arrange them in increasing order of growth. a. $10n^2 +100n +1000$ b. $10n^3 - 7$ c. $2^n + 100n^3$
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1answer
25 views

Probability of an even function

given the function: why does: does the fact that it's an even function has anything to do with it?
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1answer
21 views

Strategy for Unbalnaced Gamber Ruin

A gambler plays the following game: A fair coin is tossed until getting three times continuously head. When that happens the Gambler gets 20$\$$. Each round costs the gambler 1$ (even if he won the ...
1
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1answer
41 views

$X$ and $Y$ are ind. exponentially dist. ran. variables w/para. $\beta_1$ and $\beta_2$. Let $U=X+Y$, verify that $f_u(u)= \int_0^u f_{xy}(u-v,v)dv$.

I am a little lost with transformations with exponential distributions, any help would be much appreciated! The given hint is $0<x<infty$ and $x=u-v$
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2answers
28 views

Probability Question (Lightbulb Problem) [on hold]

This question is asking me to find the probability of non-failure of electricity in this chain. i= 1,2,3,4 Those are the probabilities of success I believe