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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Is the result of a Monte-Carlo simulation of a continuos function and with continuos input distributions again continuous?

Is the result of a Monte-Carlo simulation of a continuos function and with continuos input distributions again continuous? Suppose, we have a continuos function $f$ and a number of continuous random ...
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1answer
25 views

Probability to fit $1950$ items in a box that hold $1880$. Risk $5$%

If I buy $1950$ plates to fill a box that hold $1880$ what is the probability that $1950$ is enough if the risk of dropping a plate is $5$% per plate? The answer is $F_z(-2,81)=0,0025$ I just don't ...
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2answers
43 views

What is the probability that at least $2$ out of $4$ digits in a code is the same?

If I choose $4$ digits for a code randomly out of the digits $0$ to $9$. What is the probability that at least $2$ of these digits are the same? By at least I mean that you have to count with the ...
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1answer
19 views

Conditional Probablity for two independent events(Formula)

Let there be two independent events $A$ and $B$. To calculate the probability (for a particular condition) we have two relations. $P(A \cup B)=P(A)+P(B)-P(A \cap B)$. $P(A/B)P(B)=P(A \cap B)$, i.e., ...
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{students 1 and 2 are in different groups} vs {students 1, 2, 3, and 4 are in different groups}

Source: Example 1.11, p 26, *Introduction to Probability (1 Ed, 2002) by Bertsekas, Tsitsiklis. Hereafter abbreviate graduate students to GS and undergraduate students to UG. Example 1.11. A ...
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3answers
21 views

What is the probability that a randomly chosen positive three-digit integer is a multiple of $7$?

What is the probability that a randomly chosen positive three-digit integer is a multiple of $7$? Is my answer right?: $\frac{100}{7} = 14$, $\frac{999}{7} = 142$ Then there is $142 - 14 = 128$ ...
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0answers
11 views

Equality in Conditional Jensen's Inequality

Conditonal Jensen's Inequality says that for a convex function $\varphi$, a random variable $X$, and a sub-sigma-field $\mathcal{F}$, $E[\varphi(X)\mid \mathcal{F}] \geq \varphi(E[X\mid ...
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1answer
20 views

What is the probability of unions of intersections?

Suppose we have two unions of (possibly overlapping) events. Let me denote the unions as: $$A = IE_A^1 \cup \dots \cup IE_A^{k_A}$$ $$B = IE_B^1 \cup \dots \cup IE_B^{k_B}$$ Each $IE_X^y$ is a ...
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0answers
8 views

On the probability distribution of iterated permutations

I have this little problem that has been nagging me for a couple of months now. It occurred to me when considering the fairness of card shuffling methods. Here's my best attempt at formalizing it: ...
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0answers
27 views

Mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$?

Is the mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$? In particular, in the extreme case that the pairwise mutual informations are ...
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2answers
42 views

Modeling with Markov Chains and one-step analysis

I have set up the following model: Let $X_n$ be the number of heads in the $n$-th toss and $P(X_0=0)=1$. I can calculate the transition matrix $P$. Define $$ T=\min\{n\geq 0\mid X_n=5\}. $$ Then ...
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2answers
20 views

What is the probability or percentage or frequency by which hello line will be printed?

I have a below method which is called every one minute from background thread and that background thread keeps running always. ...
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1answer
66 views

Equivalence in conditional probability

I am wondering the equivalence of the following problem. When we computing $$ P(\mathbf{x}_1 | \mathbf{x}_2, \mathbf{x}_3) $$ is it equivalent as following, at first define $\mathbf{y} = ...
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3answers
26 views

Is every bounded sequence of random variables in $L^1$ convergent? [on hold]

If $\{X_n\}_{n>0}$ is a bounded sequence of random variables is it true that $E(X_n)$ converges?
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0answers
18 views

Destined pair 'guessing' game

n people participate in a game. Before the game the participants are put into random secret 'destined' pairs. Each round the participants pick1 their own pairs and ...
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0answers
20 views

probability,calculus

Let $N_t$ be a Poisson process and $S_{N_t}=X_1+...+X_{N_t}$. Let $A_t=t-S_{N_t}$ and $B_t=S_{N_t}-t$ 1) Show $P(B_t \geq x \ \text{and}\ A_t \geq y)=\frac{1}{E(X_1)} \int_{x+y}^{\infty} P(X_1 \geq ...
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1answer
28 views

Need Help with continuous random variable probability problem [on hold]

Suppose that an electric device has a life length $X$ which is considered as random variable with pdf: $f(x)=e^x$, $x>0$. Suppose that the cost of manufacturing one such item is $2$. The ...
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0answers
25 views

Should I use law of large numbers or Chebyshev inequality?

I think the answer is zero. Can anyone tell me whether I should use Weak Law of Large Numbers or Chebyshev inequality . I just need a hint how to proceed. Is my answer 0 correct? Thanks link to ...
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0answers
39 views

Expectation and Variance of $X/(X+Y+Z)$

I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal ...
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1answer
26 views

Continuous probability - calculate probability of r.v and distribution function

This is the question: $X$ is a continuous random variable whose probability density function is given by $$f(x)=\begin{cases} \frac{1}{9}x^2 & \text{if $0\leq x \leq 3$}.\\ 0 ...
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1answer
37 views

How to show a sequence of independent random variables do not almost surely converge by definition?

I have a sequence of independent random variables $X_1, X_2, \ldots$ where $$ X_n = \begin{cases} 1 & \quad \text{with probability} \ 1/n \\ 0 & \quad \text{with ...
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2answers
761 views

Probability of no more than three heads given that at least one toss resulted in heads

You toss a coin four times. Find the probability of no more than three heads given that at least one toss resulted in heads. So if I set event A as no more than three heads and B as at least one ...
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1answer
15 views

Mutual exclusive events are dependent

Prove that if $A$ and $B$ are mutually exclusive, $P(A \text{ and }B)=0$, then $A$ and $B$ are dependent. So I know that $A$ and $B$ are independent if $P(A \cap B)= P(A) P(B)$ How do I use ...
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1answer
34 views

What is the intuitive difference between almost sure convergence and convergence in probability?

It is a standard fact in probability that almost sure convergence is stronger than convergence in probability. I can only see the differences in the proof. However, is there a way to view it ...
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1answer
27 views

Door Prizes - Probability [on hold]

Joe goes to a party with three friends. There is a drawing for four door prizes. Each person has an equal chance of wining a prize. No one can win more than one prize. If there are totally thirty ...
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1answer
26 views

Determine how likely it is that a set of boolean data is produced by a distribution

Suppose we have a collection of independent Boolean random variables $X_i$ and $Y_i$ (for $1 \le i \le N$), and are told $p_i = P(X_i = 1)$ for all $i$. We are now given a set of values $x_i$ that was ...
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1answer
19 views

How to show convergence in probability by just using the definition?

I have a series of random variables $X_1, X_2, \ldots$ where $$ f(X_n) = \begin{cases} 1/n & \quad \text{if} \ X_n = 1 \\ 1-1/n & \quad \text{if} \ X_n = 0 \\ 0 & ...
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1answer
19 views

How can I compute the mean of a sequence of random variables?

Suppose that I have a sequence of random variables where $X_1, X_2, \ldots$ where the pdf of $X_n$ is equal to: $$ f_n(x) = \begin{cases} (n-1)/2 & \quad \ -1/n < x < 1/n \\ ...
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1answer
29 views

Continuous Probability - Bus Arriving

I am trying to do the following question: Number 24 and number 42 buses arrive independently at the corner of Mayeld Road at a random rate of 3 and 4 per hour respectively. You arrive at the ...
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1answer
15 views

Moment generating function and convergent random variables

denote by $X$ and $X_n$, $n\in \mathbb{N}$, random variables and $r\in\mathbb{R}_+$ with $E=\mathbb{E}\left[ e^{rX} \right] < \infty$ and $E_n=\mathbb{E}\left[ e^{rX_n} \right] < \infty$ for all ...
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1answer
13 views

Finding the MLE for an open interval.

So the problem says: Let $X = (X_{1},...,X_{n})$ be a random sample, where $X_{i} \sim Unif (0, \theta _{0})$, where $\theta _{0} \in (0,\infty)$ is unknown. Find the maximum likelihood estimator $T$ ...
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0answers
20 views

Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...
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0answers
23 views

Probability to get from A to C.

There has been a snowstorm and Bob is trying to drive from A to C. p and q are the probabilities that the two roads are passable. What is the probability that Bob can get from A to C? Note that ...
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0answers
36 views

Single, 6-sided die probability

I'm working on an assignment and I'm more or less new to stats. It might be the wording of the questions that's getting me as well. It deals with a regular 6-sided die. 1.a) What is the mean number ...
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20 views

poisson process(exercice)

Let $N_t$ a Poisson process and $S_{N_t}=X_1+...+X_{N_t}$. Let $A_t=t-S_{N_t}$ and $B_t=S_{N_t}-t$ 1)Show $P(B_t \geq x \ and\ A_t \geq y)=\frac{1}{E(X_1)} \int_{x+y}^{\infty} P(X_1 \geq u)du$ with ...
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0answers
31 views

Finding MLE for $\mu^{2}$

The problem says the following: Let $X = (X_{1}, ..., X_{n})$ be a random sample, where $X_{i} \sim N(\mu_{0},1)$, where $\mu_{0} \in \mathbb{R}$ is unknown. I do not have problems calculating the ...
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2answers
44 views

Defining the states when we roll one single die repeatedly

We roll a single die and the game stops as soon as the sum of two successive rolls is either 5 or 7. We want to find the probability that the game stops at a sum of 5. It seems like Markov ...
3
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1answer
45 views

How to find $z$-score

I have some probabilities, but I have to find the $z$-score. I am not sure how do to this when I am told I have to use slope-intercept. Where do I plug the numbers in exactly? Here is one of my ...
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1answer
32 views

The probability of being dealt at least 5 wanted cards

In a fictional deck of cards, there are 30 cards, 15 different ones (each card has an identical pair, so 15 pairs = 30 cards). I want to answer the question: I am dealt 10 cards. I wish to receive 5 ...
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0answers
11 views

Conditional probability,two conditions

A doctor operates on patient with a certain disease if he is 80% sure that he has it.We have a patient for whom the doctor is 60% sure that he has the disease,so he makes him do another test which is ...
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1answer
35 views

Probability of an array having all distinct numbers

Suppose you have an array of size $2n$. There are two times $2n^2$ distinct numbers that can be put into the array without replacement, i.e. for each choice of number, there are two copies, so a ...
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1answer
10 views

Test predictability with Bayes' Theorem

Say we have a disease and a test for it. P(A :=a person has the disease)= 0.01 ( example) P( B:=test is positive | A )=0.95 Is this enough information to calculate the probability that a person has ...
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4answers
16k views

Could someone explain conditional independence?

My understanding right now is that an example of conditional independence would be: If two people live in the same city, the probability that person A gets home in time for dinner, and the ...
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1answer
41 views

Urn with white and black balls, probability of n white with nth position

An urn contains $w$ white and $b$ black balls. $n$ extractions with replacement are made. What are the probabilty of: get $r$ white, $P(W=r)$ get $r$ white with white ball on n-th, n-th and n-th ...
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1answer
46 views

Show $ (\int_{-\infty}^\infty \sqrt{p}\sqrt{q}d\mu)^2\leq 2 \int_{-\infty}^\infty \min\{p,q\}d\mu $

Consider a random variable $X$ in $(\Omega, \mathcal{F}, \mathbb{P})$. Let $p,q$ be two densities with respect to a measure $\mu$ in $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where ...
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0answers
11 views

Stationary process vs stationary increments

Am I right that these are not the same, i.e. a stationary process need not have stationary increments and vice versa? example: Brownian motion is not a stationary process but it has stationary ...
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10 views

renewal process and Markov property

Let $A_t=t-S_{N_t -1}$ with $N_t$ a renewal process 1)Show $A_t$ checks the Markov property my proof: $S_{N_t}=X_1+\cdots+X_{N_t}$ and the increments are independents $$P(S_{N_t-1}=t-y\mid ...
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1answer
365 views

given lognormal distribution, find expected value of its function

I know that $Y(t)$ is a lognormal function with $$E[\log(Y(t)]=\log(Y_0)-10t$$ and $$Var(\log(Y(t))=2t$$ Given this information, how do I find $$E[(Y(t)+3)^2]?$$ I'm guessing I need to somehow use ...
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1answer
144 views

Find the Maximum Likelihood Estimator of $\theta$.

Let $X_1,X_2,...,X_n$ be a random sample of size $n$ from a population with density $f(x) = \left\{ \begin{array}{lr} e^{\theta-x} & , x \geq\theta\\ 0 & , \text{otherwise} ...
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4answers
40 views

Probability of second card being an ace

I have this task about cards: Consider choosing a card from a well-shuffled standard deck of 52 playing cards. Suppose that, after the first extraction, the card is not reinserted in ...