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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1answer
53 views

Using the symmetry assumption in this familiar probability problem

I'm revising some probability and have run into this old problem (context: Monte Carlo tests): Suppose there are random variables $t_0,t_1,\ldots,t_B$ that are independently and identically ...
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0answers
33 views

Joint transformation of a gamma distribution

I have a question regarding the transformation of a gamma distribution. I think I solved the problem, but I am not sure whether it is correct. Let $X$ and $Y$ be independent and Gamma distributed ...
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3answers
58 views

Exponential distribution wait time probability

I would like to check my answer, I have been asked to work out the probability of value greater then 10 given an exponential distribution with a mean of 10. My intuition would be that this is equal ...
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1answer
36 views

large permutation question

What are the permutations of the following: 7 marbles each of 4 colors, for a total of 28 marbles. A 5x5 board, so 25 places for 1 marble to be placed. What are the permutations of placing the 25 ...
2
votes
1answer
55 views

Variance of the sums of all combinations of a set of numbers

Let's assume a finite set of $n$ real numbers: $$\mathbb{V}=\{a,b,c,...,z\}$$ Now we take all the possible combinations of this set, including $0$: ...
2
votes
1answer
132 views

What is the moment generating function of Dirichlet distribution?

I want to find the moment generating function (or the Laplace transform) of the Dirichlet distribution. I know the moments can be found using the gamma functions as follows :$$E\left[\prod_{i=1}^K ...
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1answer
129 views

Calculate the expected value of the following betting strategy.

I am trying to calculate the expected value of the following game: we first start off with ten 10 in a row. Now the probability to win is $p(w)$ and the probability to lose is $p(l)=1-p(w)$. If we ...
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0answers
49 views

Misprints in a book as Poisson process

Suppose that a book with $n$ pages contains on the average $λ$ misprints per page. What is the probability that there will be at least $m$ pages which contain more than $k$ misprints ? I am getting ...
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1answer
88 views

Coefficient of variation of an hyperexponential

This question is on my mind for days and I haven't find the answer. Can someone help me? Suppose there exists a super awesome hyperexponential random variable $X$ with $k$ exponential variables with ...
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2answers
56 views

Why is this wrong - conditional expectation of brownian motion: $\mathbb{E}[B_1 | B_2]$

Trying to find $$ \mathbb{E}[B_1 | B_2]$$ $$\mathbb{E}[(B_1 - B_2 + B_2) | B_1] = \mathbb{E}[(B_1-B_2)|B_2] + \mathbb{E}[B_2 | B_2] $$ $$\mathbb{E}[ -(B_2 - B_1)| B_2] + B_2$$ Since $$ -(B_2 - ...
0
votes
1answer
25 views

Player's View: Probability of number of certain die on table given dice in hand

I'm trying to make some AI for a single player version of a dice game named Dudo. The relevant aspects are that there are six players with six dice each (which only they can see until the end reveal), ...
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0answers
83 views

Confusion about a random process

Let $X(t)$ be a random process such that: $$ X(t) = \begin{cases} t & \text{with probability } \frac{1}{2} \\ 2-at & \text{with probability } \frac{1}{2} \\ \end{cases}, $$ where $a$ is a ...
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0answers
36 views

Number of Data Samples Needed for 80% Confidence Level for Flight Test?

Simply trying to determine the minimum number of times to repeat a test that will give us an 80% confidence level that Mode A is tracking better than Mode B. Each test will involve an aircraft sensor ...
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0answers
53 views

Heteroskesdacity

Consider the following model for real estate values applied to a cross-section of homes: $Price = \beta_0 + \beta_1\cdot SQFT_i + \beta_2 \cdot YARD_i + \beta_3 \cdot POOL_i + \epsilon_i$ where ...
1
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1answer
77 views

You have to draw 10 cards. What is the probability that you will draw at least one repeated card? [duplicate]

Assume you have 52 cards in a deck, and you have to draw 10 cards. Assuming you have to put back the card after each draw, what is the probability that you will draw at least one repeated card? ...
2
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0answers
34 views

Under what conditions on the experiment does bootstrapping work?

For a proof I would like to pretend that the uniform distribution on a finite set of samples from a 'source' eventually becomes the source's distribution a.s. when you keep adding samples. I am not ...
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votes
2answers
92 views

What is the probability of 12 friends each having birthday on different months of each other. [closed]

What is the probability of 12 friends each having birthday on different months of each other. Twelve people having birthday on every month of the year. We are a group of 12 friends and each one of ...
0
votes
1answer
24 views

Using binomial distribution to find probability range

If the probability that a moose has three antlers is z and I have 500 moose, I want to find the probability that more than 8 moose have three antlers. Since these are independent events, I want to use ...
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2answers
74 views

Double exponential function

Suppose $X$ has an exponential distribution with parameter $1$ and $Y=ln(X)$. The distribution of $X$ will be $f(x)=e^{-x}$. Now I want to find the distribution of $Y$. So say: ...
20
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3answers
12k views

Expectation of the min of two independent random variables?

How do you compute the minimum of two independent random variables in the general case ? In the particular case there would be two uniforms variables with difference support, how should one proceed ? ...
1
vote
1answer
44 views

On $[0,1] $ $, 100$ points are chosen at random. $X_1$- number of chosen point between $\frac{1}{5}$

On $[0,1] $ $, 100$ points are chosen at random.(This probably means, uniformly, I can only assume, no other context is given, so let's pressume what seems most natural.) $X_1$- number of chosen ...
1
vote
1answer
51 views

Number of possible cases on RNA strand.

A RNA strand made up of $22$ base pairs can have A, C, G or U per each pair. If the base C are continuously arranged for $4$ times like AGUGCCCCAAAAAAAAAAAAUU this RNA strand could act as a blueprint ...
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1answer
46 views

Distribution function?

Let $F(x) = e^{-1/x}$ for $x>0$ and $F(x)=0$ for $x\leq0$. Now I am investigating if $F$ is a distribution function. Say: \begin{align} \int\limits_0^\infty e^{-1/x} \, dx = \left[ ...
0
votes
1answer
50 views

Horse race - once vs. twice

You select your horse for a race. The horse has a parameter $q \in [-0.3,+0.3]$. Its probability of winning depends on the weather, which can be either Rainy or Sunny: In Rainy weather, the ...
2
votes
1answer
57 views

20 balls are put into 10 boxes, let $X$ be the random variable that accounts for the number of empty boxes. Find $EX$ and $DX$-variance.

What I don't know how to do: Put this into a mathematical model effieciently, I honestly do not know where to start here, I've done problems with balls going into boxes and to find lets say the ...
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3answers
65 views

On the interval $(0,1)$ $9$ points are chosen at random.Let $X$ be the distance from $0$ to the medium between the chosen points.

On the interval $(0,1)$ 9 points are chosen at random.- This most likely means uniformly, I doesn't say more than was is written, just the presumption of choosing these points is what comes to mind ...
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0answers
20 views

Bayesian posterior with a constraining equation (slice-projection?)

Prior and signals: Let $y_1$ and $y_2$ be iid signals on $Y$. The intial prior is $Y \sim N(\bar{Y}, \sigma^2_Y)$, where $N(\cdot, \cdot)$ is the normal distribution The signals are independent and ...
0
votes
1answer
54 views

Limit of the normal density function as the variance and the mean approache zero

$f(v,x)=\frac{1}{\sqrt{2\pi}v}\exp(-\frac{1}{2}(\frac{x}{v}-1)^2), x \ne 0, v >0$ $f(v,x)$ is the density of a normally distributed r.v. with mean $v$ and variance $v^2$. I need to show that ...
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0answers
144 views

Compute (a) P(X = 2|Y = 3) and (b) P(Y = 3|X = 3) for the following joint distribution

I have some trouble understanding a question from my testbook. The question is as follows: Compute (a) P(X = 2|Y = 3) and (b) P(Y = 3|X = 3) for the following joint distribution: Y X=1 2 3 1 ...
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2answers
88 views

Filtration of Markov Chains in general state space

I am reading the book Markov Chains and Stochastic Stability from Meyn and Tweedie. They define Markov chains on a measurable state space $(E,\Sigma)$ (Chapter 3.4) and they define it on the space ...
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2answers
46 views

Let $X$ be a continuous random variable with density function $f_X$. What is $Y=aX+b$?

Let $X$ be a continuous random variable with density function $f_X$ and let $a,b>0$. What is $Y=aX+b$? I need some help with this one. And I am quite sure it is not $af_X+b$.
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0answers
64 views

Differentiate a gaussian process

Let $(X_t)_{t\in [0,1]}$ a centered Gaussian process. Assume the covariance function $K$ to be $\mathcal{C}^2$. Show that for all $t\in [0,1]$, $$\tilde{X}_t=\lim _{h\to 0}\frac{X_{t+h}-X_t}{h}$$ ...
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0answers
35 views

Probability of a sample mean from a continuous distribution

Simple question, which I'm pretty confident my teacher is conveying incorrectly: If I have a continuous random variable, and take a finite sample from it (say, n=50), what is the probability of any ...
2
votes
1answer
60 views

Show that if $P(0 \leq X \leq c)=1$ then $Var(X) \leq \frac{c^2}{4}$

I need to show that if $$P(0 \leq X \leq c)=1$$ then $$Var(X) \leq \frac{c^2}{4}$$ I can show that using 2 things: First, that $E[X^2] \leq cE[X]$ and secondly that $Var(X) \leq ...
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1answer
520 views

Calculating the probability that the demand will exceed what we have in stock

You identified two products that have common average weekly demand, but different standard deviations. Product 1's weekly demand is distributed normally with a mean of 625 & standard deviation ...
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0answers
23 views

Sample variance matlab geometric brownion motion

I have a question about the geometric Brownian motion. I want to sample many paths and then showing that the sample variance equals the exact variance: $$\mathrm{Var}\left[S(t)\right]=S_{0}^2 e^{2 \mu ...
0
votes
1answer
65 views

Finding joint pdf of two random variables. [closed]

We are given the probability distribution function (PDF) of the random variable $X$ $f_X (x) = 0.5 \big( u(x+1)-u(x-1) \big)$ and another random variable $Y$ is given as $Y=X^2$. We can find the PDF ...
0
votes
3answers
1k views

Rolling 6 dice and 4 on the same side

Guys I made myself a question to answer and its this. What is the probability of having 4 dice land on the same side if I toss 6 dice. I got this answer, I don't know if its correct. I dont want to ...
0
votes
1answer
28 views

Why is the co-variance multiplied by two when summing the variances of two correlated random variables?

As I understand, when trying to sum the variances of two correlated random variables, the formula is such: $Var(X+Y) = Var(X) + Var(Y) + 2Cov(X, Y)$ My question is, why is the last term multiplied ...
0
votes
1answer
57 views

Writing the conditional expectation of a function of a random variable in a different way

Let $p_1$ and $p_0$ be two probability density functions and let the function defined by $$l(y)=\frac{p_1(y)}{p_0(y)}$$ be the probability ratio function. Random variable $Y$ has a density function ...
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4answers
349 views

Use a 20 sided die and a 6 sided die to represent number 1-45 with equal probability

I wonder if anyone could help me with this question: Use a 20 sided die and a 6 sided die to represent all numbers from 1-45 with equal probability. Also, it is okay to use one die repeatedly as ...
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2answers
153 views

You have to draw 10 cards. What is the probability you will not drawn any repeated cards?

Assume you have 52 cards in a deck, and you have to draw 10 cards. Assuming you have to put back the card after each draw, what is the probability that you will not draw any repeated cards? Here's ...
2
votes
1answer
191 views

Probability of people in a queue

Here is a simple model of a queue. The queue runs in discrete time (t = 0, 1, 2, . . .), and at each unit of time the first person in the queue is served with probability p and, independently, a new ...
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0answers
28 views

What is the axiomatic definition of potential kernel?

In the enter link description here, a kernel V is a Markov potential kernel if the operator I+V is invertible and its inverse is of the form I􀀀-N with N a sub-Markov kernel.The definition of Markov ...
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2answers
261 views

Probability of Opening a Combination Lock

A combination padlock has three rotating disks each with numbers 1 through 9. You forgot your code but remember that the sum of the three numbers is 8. In an attempt to open the lock, you begin ...
2
votes
1answer
221 views

Expected Value and Loops

I have the following question: Imagine you have $N$ pieces of rope in a bucket. You reach in and grab one end-piece, then reach in and grab another end-piece, and tie those two together. What is the ...
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votes
2answers
46 views

How am I misusing the Bayes' rule?

I just started studying probability. I am trying to solve this exercise: When coin A is flipped it comes up heads with probability 1/4, whereas when coin B is flipped it comes up heads with ...
2
votes
2answers
23 views

Fast way to calculate sequence

is there a fast way to calculate the sequence: $f_k = 0.5 * (f_{k-1}+1) + 0.5* (\frac{1}{f_{k-1}})$ for $f_7$ with $f_1=100$? Specifically, the question was that a coin was thrown: If I get heads, ...
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1answer
61 views

Probability Distribution Function $W=X+Y$ [closed]

$(X,Y) \sim f(x,y)=2x \space (0 \leq x \leq 1, |y| < x^2)$ $W = X+Y$ pdf $f_w(w)$ how do I get the value $f_w(w)$ ? Is the span divided ? I am not sure .... T.T I think x,y are not ...
0
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0answers
58 views

Prove that $\mathbb{P}[(\limsup A_n)-A_n]\rightarrow 0$

Problem: This is a problem from Problems in Probability by Shiryaev (Problem 2.1.16). It is worded as following: Let $A^*=\limsup A_n$. Prove that $\mathbb{P}(A^* -A_n)\rightarrow 0$ Attempt: I tried ...