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
19 views

Probability question - as91685

If 3 students who own cell phones are selected at random, calculate the probability that two have smart phones and one has a non-smart phone. State any assumption you make in calculating your result ...
3
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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
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0answers
20 views

Compute probability $P1=\sum_{j=n+1}^{2n+1} …$

I have a following problem: A school has $N$ students in which $M$ students are leader (of each class in school), and $N>M$. There are $2n+1$ balls in the black box including $n+1$ blue balls and ...
0
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2answers
47 views

Probability query

8 Indians and 3 Americans are to stand in a row at random. What is the probability that no 2 Americans are always together? I think the answer should be 1 because as there are many possible ...
0
votes
1answer
32 views

Mixing continuous and discrete distributions

I'm wondering how, if it is at all possible, to write the p.d.f. for the following random variable. Given RVs $X_1$ and $X_2$ distributed according to some joint distribution having known density ...
2
votes
1answer
53 views

Sufficient statistic

Let $\mathbf{X}=(X_1,\ldots,X_n)$ with joint frequency function $f(\mathbf{x};\theta_1,\theta_2)$ where $\theta_1,\theta_2$ vary independently. The set ...
2
votes
1answer
19 views

Minimum number of samples to take so that proportion of smokers in sample is within a certain threshold?

What is the minimum number of random samples that should be taken so that with probability at least 0.95, the proportion of smokers in the sample will not differ from the unknown population of smokers ...
4
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2answers
38 views

Covariance of $X^2$ and $X^3$ when $X$ is exponentially distributed

Here is my work.... $\begin{align*} Cov(Y,Z) &= E(YZ) - E(Y)E(Z)\\ &= E(X^2\cdot X^3) - E(X^2)E(X^3)\\ &= E(X^5) - E(X^2)E(X^3) \end{align*}$ And we know $E(X^n) = ...
0
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0answers
23 views

Prove Joint distribution of estimators

Let $X_1,...,X_n$ iid r.v. with distribution F, with mean $\mu$ and median $\theta$.Assume that $Var(X_i)=\sigma^2$ and $F'(\theta)>0$. If $\hat{\mu}_n$ is the sample mean, and $\hat{\theta}_n$ the ...
2
votes
1answer
53 views

Intuition in probability theory

Good afternoon. Could you please suggest me some books or may be articles where I can read about the intuition of Kolmogorov's axiomatics. I know it, I can solve university problems but I can't feel ...
6
votes
3answers
113 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
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2answers
23 views

Probability of Selecting 3 Letters from 7 choices

$A,B,C,D,E,F,G$ A list consists of all possible three-letter arrangements formed by using the letters above such that the first letter is $D$ and one of the remaining letters is $A$. If no letter is ...
1
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2answers
31 views

Asymptotic conditional distribution of normal variable

$X$ is a normal variable $\mathcal{N}(0,1)$, $Y$ is a normal variable $\mathcal{N}(n,n-1)$, independent of $X$. I want to prove that the distribution of $X$ conditionally on $X > Y$ is ...
1
vote
1answer
22 views

Deriving a lower bound for a probability involving a random variable $X$ with the Gamma distribution.

Question Let $X$ have the $Gamma(\alpha, \beta)$ density. I.e. $$f_X(x) = \frac{1}{\gamma(\alpha)\beta^\alpha}x^{\alpha-1}e^{-\frac{x}{\beta}}$$ when $x >0$ and $0$ elsewhere. The moment ...
0
votes
1answer
64 views

loi du quotient de variable aleatoire [on hold]

Salut, J'ai un souci si je considere une variable X qui suit une loi de cauchy et une autre Y qui suit une loi normale centree reduite. Quelle est la loi de X/Y? Et si X suit une loi centree reduite ...
0
votes
1answer
22 views

Multivariate distribution

Let $X_1, X_2, X_3$ be independent random variables with normal distribution $n(0, \sigma^2)$. Let $Y = (X_1^2 + X_2^2 + X_3^2)^{1/2}$. Find the density of $Y$. I tried finding the densities of ...
1
vote
1answer
51 views

Basketball shots and stopping rule

You are taken to play a basketball game where you can shoot basketballs at n slots using a machine that is equally likely to shoot the balls into those n slots. You can stop whenever you see fit and ...
0
votes
2answers
23 views

Probability density use for biased outcome

I am not a mathematics pro so do not mind if this is dumb let us suppose I have a method for generating random real values between 0 and 1 . All the values between 0 and 1 are equally likely to be ...
0
votes
0answers
13 views

Entropy of the product of two random variables

Consider a random matrix $X$ and a random vector $Y$. Let the Shannon entropies $H(X) = H(Y) = n$. Is there a simple upper bound for entropy $H(XY)$? I believe $H(XY) \leq 2n$ as that is a simple ...
0
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0answers
13 views

Q: Finding probability of connection based on distance?

So, I am new to graph theory and statistics but have encountered a problem that I am not exactly sure how to solve. I have a graph with n nodes and am trying to determine the probability of connection ...
0
votes
1answer
36 views

Simple Expected value of MLE

Let $X_1,..., X_n$ be iid $Exp(\lambda)$. The MLE for $\lambda$ is $\hat{\lambda}=\frac{1}{\bar{X}}$, where $\bar{X}=1/n \sum^n_{i=1}X_i$ How can I conclude that $E(\hat{\lambda}) = n\lambda/(n-1)$? ...
1
vote
1answer
26 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
0answers
13 views

recurrence simple random walk in one dimension before hitting time

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
19 views

Proving that a process has the Markov property

Let $X_t=xe^{ct+aB_t}$ where $B_t$ is one dimensional Brownian motion. How would I prove this is a Markov process using the expectation definition of a Markov process, i.e., ...
1
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2answers
49 views

Efron-Stein inequality

The Efron-Stein inequality sais that if $X_1,\ldots,X_n$ are independent random variables on say $R^n$, and $f:R^n \rightarrow R$ s.t. $Z:=f(X_1,\ldots,X_n)$ has finite variance, then ...
4
votes
1answer
45 views

Probability of drawing a run of a specific color from an urn with two colors of balls

I was sent a puzzle involving an urn with 128 white balls and 288 black. If the balls are drawn without replacement until the urn is exhausted, what is the probability that a sequence of 10 or more ...
1
vote
1answer
36 views

$E(X_i \cdot I(X_i>\theta)$ expected value of when X is greater than the median.

Let $X_1, ..., X_n$ be iid with a distribution F. Let $\theta$ be the median of F. What is the value of $E(X_i \cdot I(X_j>\theta))$? If $i\neq j$, then $E(X_i \cdot I(X_j>\theta))= 1/2 \cdot ...
1
vote
1answer
33 views

Random walks with finite chance of escape

In a recent answer I gave a combinatorial interpretation for the sum $\sum_{n=1} \binom{2n}{n}\frac{4^{-n}}{n+1}=1$: namely, that it corresponded to the probability of all outcomes adding to $1$. A ...
0
votes
1answer
12 views

Histogram with different sample probabilities

Assume we are given a list of samples $L_1,L_2,\ldots,L_n$ of some random variable $L$. By classing them into bins we can easily create a standard histogram. But now suppose that we associate a ...
1
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2answers
31 views

what is the > probability that only one letter will be put into the envelope with > its correct address?

Tanya prepared 4 different letters to 4 different addresses. For each letter, she prepared one envelope with its correct address. If the 4 letters are to be put into the four envelopes at ...
1
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0answers
17 views

Working with the sum of two independent random variables, and estimating a parameter

A network source sends a sequence of zeros and ones, $X_1, X_2, ...$ with $X_i$(iid) Bernoulli with $p = P(X_i = 1), 0 < p < 1$. Due to disturbances the received sequence is $Y_1, Y_2, ...$ ...
1
vote
1answer
16 views

How do I combine probability estimates of two equivalent/mutually inclusive events?

Let's say I'm pregnant with fraternal twins. One of them hangs out in the left side of my womb, and the other hangs out on the right (I have no idea how biology works). We've applied a flaky test to ...
4
votes
1answer
65 views

Why if we use independence and factorization, we cannot represent every joint distribution? (rigorous argument needed)

I was reading Koller's Probabilistic Graphical models book and it says something like this: Let $P(x_i) = \theta_i$. Define: $$P(x_1, \ldots , x_n) = \prod_{i=1}^n \theta_i$$ This ...
4
votes
3answers
85 views

In 30 boxes are 15 balls. Chance all balls in 10 or less boxes?

Question1: I found 30 boxes. In 10 boxes i found 15 balls. In 20 boxes i found 0 balls. Afer i collected all 15 balls i put them randomly inside the boxes. How much is the chance that all balls are ...
0
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0answers
13 views

Simulation Lévy process

I need to simulate a Lévy process from its characteristic triple $(\gamma,\Sigma,\nu)$ where $\nu$ is the Lévy measure. I know that I can simulate it by summing a brownian motion and a compound ...
0
votes
1answer
35 views

Mean absorption time for pure birth process

Let $\xi_t$, $t\geq0$, be a pure birth process, with $P\{\xi_{t+h} = i +1 | \xi_t = i\} = \lambda^ih + o(\lambda)$, as $h \downarrow 0$. At $t=0$, $\xi_0 =1$. Let $\tau = \min\{t ~|~ \xi_t = N\}$. ...
0
votes
1answer
36 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
1answer
33 views

The probability of drawing at least two diamonds among three cards drawn at random with replacement

I am learning Random variables and Probability distribution. I got this question some what hard! Can somebody help me solve this please. Three cards are drawn at random successively with ...
0
votes
1answer
78 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 ...
2
votes
2answers
59 views

Why is this expectation true?

Working with Rao-Blackwell, this came up: $$E[2X_1 \mid \max(X_i) = t] = 2\left(\frac{1}{n}t + \frac{n-1}{n}\frac t 2\right)$$ Where X are uniform(0, $\theta$). What are the intermediate steps? I'm ...
-4
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0answers
23 views

Probability of two people calling the same person at the same time [on hold]

I have a smartphone, and a few days ago, I somehow called someone in my contacts while my smarthpone was in my back pocket while standing up, at the same time a person I was talking to called them. ...
4
votes
2answers
64 views

Probability for Magic Trick

I had a probability problem to solve , but could not proceed further , we have a m identical decks having n cards , where each deck has n different cards . Now shuffle them and select n cards . Now a ...
1
vote
1answer
20 views

Expected value of $(\overline{X} - 5)^2$?

$X_1, \ldots, X_9$ are 9 random samples from a N(5, 9). I am looking for the distribution ofExpected value of $(\overline{X} - 5)^2$. $$E[(\overline{X} - 5)^2] = E[\overline{X}^2 - 10\overline{X} + ...
2
votes
2answers
1k views

Two cards are chosen from a deck of 52 cards without replacement.

Two cards are chosen from a deck of 52 cards without replacement. Determine the probability that both cards are face cards or both cards are hearts? I did face cards $\frac{12}{52} \times ...
1
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0answers
10 views

Testing hypothesis about window non-overlap

I have a large number (~1.5 million) of protein sequences, each of them of different lengths.There are 6 schematic examples in the attached image. Within each of these sequences, there are >= 0 ...
1
vote
2answers
289 views

How many white balls are there in the box most probably?

There are n balls in a box. Some of them are white. A ball drawn from the box turns out to be white. How many white balls are there in the box most probably? Alright, well i know there are originally ...
2
votes
1answer
37 views

Finding conditional distribution

Let $X$ and $Y$ be independent $Exp(1)$-distributed random variables. Find the conditional distribution of $X$ given that $X + Y = c$ ($c$ is a positive constant). this is my idea: $$f_{X \mid ...
1
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2answers
39 views

Probability of Dialing Correct Digits

Jason remembers only the first five digits of a seven-digit phone number, but he is sure that neither of the last two digits is zero. If he dials the first five digits, and then dials two more digits, ...
0
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0answers
16 views

Expected number of dice rolls to uniquely determine values of the faces

Suppose we have $k$ fair $n$-sided die (so each face rolls with probability $1/n$), with each face of the die labelled as $a_i$ $ (i=1, \ldots, n)$, with the values of the $a_i$ being unknown to us. ...
2
votes
0answers
29 views

What does the notation $\textbf{x} \langle\textbf{Y} \rangle $ mean if $\textbf{Y} \subseteq \textbf{X}$ for random variables?

I was reading daphne's Probabilistic Graphical Models book and she introduces some notation about sets of random variables that I am confused about (on page 21 section 2.1.3.2). Before I ask my ...