This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under ...

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29 views

Probability a blackjack dealer will bust if you know their score and know the exact deck?

If you know the exact cards left in a deck, and the score of the dealer, how can you calculate the exact probability that they will bust? The dealer behaves as follows: If the dealer's score is less ...
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2answers
24 views

Prove that $X,Y$ are independent iff the characteristic function of $(X,Y)$ equals the product of the characteristic functions of $X$ and $Y$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $X$ and $Y$ be random variables on $(\Omega,\mathcal A,\operatorname P)$ with values in $\mathbb{R}^m$ and $\mathbb{R}^n$, ...
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2answers
50 views

How much would you pay to pull a ball from the bag?

If there are 9 white balls and one black ball in a bag. The white balls are valued at 10 dollars and black ball at 100 dollars. How much are you willing to pay for each pull from the bag (only one ...
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1answer
18 views

How to construct a transition matrix?

I'm giving my first steps in stochastic processes but I'm having some difficulties. See the following example Suppose that whether or not it rains today depends on previous weather conditions ...
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2answers
79 views

Game of probability

n a game, played between $2$ players there is a circular field and one of the players is blindfolded, who stands in the center of the field. The other player stands at a fixed point on the ...
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1answer
29 views

Clarifying Derivation of Entropy

I'm learning about probability from the book Pattern Recognition and Machine Learning by Christopher Bishop. It includes a justification for the definition of entropy that can be summarized as: let ...
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0answers
28 views

Absolute value of a sum of non-identically distributed RVs [on hold]

Let $X=\left|\sum _{i=1}^n Z_{i} \right|$ where random variables $(\textit{Z${}_{i}$})$ are independent but $not$ identically distributed, and, $Z_{i} =0$,$+1$ or$-1$, with probability $1-a_i$, ...
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0answers
21 views

Is this martingale identicaly zero?

Let $X_t$ be such that $X_t$ is bounded continuous martingale adapted to the filtration $\mathcal{F}_t$ such that $$\Bbb{E} \bigg[\int_0^t e^{X_s} \, d\langle X\rangle_s\bigg] = 0$$ Does it follow ...
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0answers
38 views

finding the limit for a martingale

I have trouble finding the exact limit for a martingale: Let $\{\xi_n\}_{n\in\mathbb{N}}$ be a Markov chain with $\xi_0$ uniformly distributed in $[0,1]$ and $$ ...
9
votes
2answers
361 views

How far do I need to drive to find an empty parking spot?

A parking lot consists of an infinite row of bays. Cars arrive at random intervals (mean interval $T_a$) and stay for a random time (mean stay $T_s$). The time intervals are memoryless (negative ...
1
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1answer
64 views

Why does this expectation integrate to 1

Let $p(y|\theta )$ be our likelihood, and $\hat{p}_{N}(y|\theta)$ be an unbiased estimator of our likelihood. Let $z=\ln \hat{p}_{N}(y|\theta) - \ln p(y|\theta )$, and $g_{N}(z|\theta)$ be the ...
0
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1answer
17 views

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Prove for any set of integers $k\leq l<m$ that

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Prove for any set of integers $k\leq l<m$ that the difference $X_m-X_l$ is uncorrelated with $X_k$, that is, $$E[(X_m-X_l)X_k]=0.$$ ...
1
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1answer
31 views

Prove that $\tilde{X}_{\tilde{\theta}}(t)$ is a martingale

Let me introduce the objects: 0) $(\Omega, \mathcal{F},\Bbb{P})$ is a probability space 1)$S_N $ is the set of symmetric, non-negative definite $N\times N$ matrices 2)$a:[0, \infty) \times \Omega ...
-1
votes
2answers
48 views

Why birthday distribution is not uniform.

I was reading about birthday problem and I found a statement that real-life birthday distributions are not uniform since not all dates are equally likely (last line ...
63
votes
9answers
11k views

If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?

Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair? Furthermore, ...
3
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2answers
175 views

Concentration inequality for the median

Most concentration inequalities talk about deviation of the sample mean from the population mean. Is there a result bounding the probability of deviation of the sample median from the median of the ...
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0answers
43 views

Conditional Expected Value of Product of Normal and Log Normal Distribution

EDIT: SIMPLIFIED EXPRESSION INCLUDED NOW Could someone please provide the answer and steps to solve this expression? \begin{eqnarray*} & & ...
2
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2answers
41 views

Probability of the simultaneous failure of two components where one component would take the full load if only one failed instead.

Working through an example question in Applied Probability for Engineers and Scientists, 1st Ed., by Ephraim Suhir. Example 1.9, beginning on p. 5, reads as follows: A heavy, nondeformable beam ...
3
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3answers
153 views

How to understand independence of probability?

By definition, when $$P(E\,|\,F) = P(E)$$ holds, we say that $E$ is independent of $F$. By definition of conditional probability, $$P(E\,|\,F) = {P(E \cap F) \over P(F)} \Rightarrow P(E \cap F) = ...
1
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1answer
45 views

Is this a misuse of the term “probability space”?

Let me first state the definitions as I am using them. Do correct me if I am wrong here! A "probability space" is a triple $(\Omega, F \subseteq 2^{\Omega}, \mu : F \rightarrow [0,1])$. The ...
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0answers
17 views

Probability of a decay process [on hold]

Consider a decay source with decay constant $\lambda$ (exponential decay) arriving with rate $R$ (poisson process). What is the probability that the product comes from the source right before it? (A ...
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2answers
32 views

Confused by certain interpretation of expected value…

I read the following in Stein / Shakarchi's Fourier Analysis book, where they discussed the notion of expectation of a probility density. "Consider the simpler (idealized) situation where we are ...
0
votes
2answers
28 views

Find the density of their average

If $f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$ find the density of their average $\frac{X+Y+Z}{3}$ I'm a little lost on how to solve this exercise, ...
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2answers
21 views

Calculate dependent probabilities

Imagine that two raffles will happen, every raffle will reward 1 person. 10 people will participate. The first raffle will reward 1 person. The second raffle will reward another person, but the ...
2
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0answers
20 views

Bayes theorem with multiple conditions

how to calculate P(a|b,c,d). knowing that b, c and d are NOT independent from each others ? i know how to solve it if there is independency assumption. however, i am just wondering if there is any ...
2
votes
1answer
21 views

Show that $|F_{X,Y}(x,y)|^2\leq F_X(x)F_Y(y)$

Consider the random variables $X$ and $Y$ defined in the same space $\Omega$. Show that $$|F_{X,Y}(x,y)|^2\leq F_X(x)F_Y(y)$$ This question comes from an old test, I know that ...
1
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1answer
20 views

The number of distinct values taken by a sequence of partial sums of iid

I'm working on an old exam problem as follows: Let $\{X_n\}_{n\in\mathbb{N}}$ be a sequence of i.i.d taking values in $\mathbb{Z}$. Define $S_0 :=0$ and $S_n:=X_1+X_2+...+X_n,$ and $\theta_n$ to be ...
1
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1answer
28 views

Find the density

Suppose that radius $R$ of one sphere is a continuous random variable with density $$f_R(r)=6r(1-r) I_{[0,1]}(r)$$ Find $f_V(v)$ and $f_S(s)$ the densities of volume and surface area I did ...
1
vote
1answer
30 views

Expectation of a random variable in terms of its distribution function

Here is a theorem on expectation of a random variable in terms of its distribution function Theroem: Let $X$ be a (continuous or discrete) non-negative random variable with distribution function ...
2
votes
2answers
48 views

Proving that the Poisson compound process has independent increments

Let $X_t=\sum_{i=1}^{N_t}J_i$ be a compound Poisson Process, where $J_i$ are independent and equidistributed. I have to prove that for every $0<t_1<t_2 \leq t_3<t_4$ : $X_{t_4}-X_{t_3}$ is ...
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2answers
34 views

A confusing probability question..

A magician holds one six-sided die in his left hand and two in his right. What is the probability the number on the dice in his left hand is greater than the sum of the dice in his right?
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votes
1answer
31 views

How do I determine the weight to assign to each bucket?

Someone will answer a series of questions and will mark each important (I), very important (V), or extremely important (E). I'll then match their answers with answers given by everyone else, compute ...
2
votes
1answer
33 views

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are ...
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2answers
33 views

Distribution of Summation of two discrete random variables

Here, $\tilde{x}_1$ and $\tilde{x}_2$ are two non-negative independent discrete integer-valued random variable and the support set of $\tilde{x}_1$ and $\tilde{x}_2$ are below: $$ X_{1} = \{ ...
1
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3answers
54 views

Probability that an integer is divisible by $8$ [on hold]

If $n$ is an integer from $1$ to $96$ (inclusive), what is the probability that $n(n+1)(n+2)$ is divisible by 8?
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votes
1answer
52 views

expected number of steps for chossing randomly each number between 1 to $n$ at least $k$ times [on hold]

Assume the following game: Every step choose a number between 1 to $n$ randomly i.e. every integer between 1 to $n$ is chosen with probability $\frac{1}{n}$. Success is when every number has been ...
1
vote
1answer
485 views

How to prove convergence in mean implies uniform integrability?

My class notes and wikipedia both say that $X_n \xrightarrow{L^1} X$ $\Leftrightarrow \; X_n \xrightarrow{P} X$ and $X_n$ are uniformly integrable. I am trying to work through the proof. I am able ...
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votes
1answer
25 views

Poisson Distribution practice problem

I have just started to learn Poisson Distribution and I have no idea how to deal with the following practice from my textbook: Suppose the average amount of cars passing on a street per minute is ...
3
votes
1answer
55 views

Why can we consider the Brownian motion as being a mapping into the space of continuous functions, even though its paths are only a.s. continuous?

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal{A},\operatorname{P})$. By definition of $B$, for $\operatorname{P}$-almost every $\omega\in\Omega$ ...
2
votes
1answer
30 views

(Elementary) Markov property of the Brownian motion

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
0
votes
1answer
30 views

Proof the statements

Proof the statements below i)If $P(A)=0$ and $B$ is any event, then $A$ and $B$ are independents ii)If $P(A)=1$ and $B$ is any event, then $A$ and $B$ are independents iii)The events ...
0
votes
2answers
369 views

Equal Chance Probability, Different Numerical Values — Average is the probability? How to beat the odds?

A person is gambling. Person has an equal chance to roll on 1 through 50. Each roll equals the amount they get (rolling a 45 will get you 45 dollars). I have no trouble figuring out the average. ...
1
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1answer
34 views

Does this game have infinite expected payout?

Consider the following game: Suppose the initial value of the pot is $ S $. Our player Josephine then rolls a fair $n$-sided die. If the roll is not $1$, then the pot is multiplied by that roll, and ...
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0answers
20 views

Probability of selecting a sequence in order

If "X" number of attempts are made by "Z" number of persons to select a random number from a range "r", where "X <= r". Then I am interested in finding the probability that a particular sequence in ...
-1
votes
1answer
34 views

Dice probability [on hold]

You roll a fair dice twice A. What is the probability that the first roll is odd and the second roll is even? B. What is the probability that one roll will be odd and the other roll will be even?
2
votes
1answer
28 views

Inequality for the derivative of a density of a random variable convolved with a normal r.v.

I have a question about the following proof. The statement is: Let $X$ be a random variable and $Z_\tau \sim N(0,\tau)$ be an independent random variable. Then $Y_\tau := X + Z_\tau$ has a ...
2
votes
3answers
8k views

What is the expected number of trials until x successes?

This is barely a probability question, but I needed to check to make sure the solution is as simple as I believe it to be. What is the the expected number $n$ of independent trials needed to have $x$ ...
2
votes
2answers
51 views

Disjoint events

Let $A$ and $B$ two disjoint events such that $P(A)=0.3$ and $P(B)=0.5$. Find the probability that i)$A$ or $B$ ocurrs ii)$A$ occur but not $B$ iii)repeat $i)$ and $ii)$ with $A$ ...
0
votes
2answers
64 views

Baye's Theorem Conditional Probability with multiple conditions

Lets assume I have a supermarket and I track the purchase history of my customers with 2 attributes of each customer - Gender (M/F) & Smiling (Y/N). Assume this is historical data of purchases: ...
1
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1answer
22 views

Proving Properties of Discrete Time Markov Chain mathematically

I want to prove that the queue length at a store is not a Discrete Parameter Markov Chain (DPMC). Now I have the equation: $$Q_k = (Q_{k-1} - 1) + V_k$$ $Q_k$ is the queue length at time instant ...