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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20 views

Conditional probability problem at Computer Science

I hopefully someone can help me with this problem of conditional probability: "A disk server receives requests from many client machines and requires 10 milliseconds to respond to each request. The ...
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2answers
13 views

Let Y be a random variable denoting the age at which a piece of equipment fails…

Let Y be a random variable denoting the age at which a piece of equipment fails. In reliability theory, the probability that an item fails at time y given that it has survived until time y is called ...
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1answer
15 views

Chance of winning in a raffle

A raffle consists of 10 sheets with 10 numbers (1 to 10) on each sheet i.e. 100 chances in total. The draw is done by first selecting a sheet at random and then selecting the winning number out of the ...
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13 views

Time-based probability question

Two adult male baboons are introduced to the same 50 ft. square cage. Male A looks at Male B for a total of 5 hours in the first day, and Male B looks at Male A for a total of 3 hours in the first ...
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1answer
13 views

A Question on the Scaling Invariance of Brownian Motion

I read the following paragraph. Let $B_t, \ t \in [0, \infty)$ be a standard linear Brownian motion. For each $q > 4$, define the following sequence of sets. $$ \Omega_k := \left\{\omega \in ...
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12 views

Probability of Multiple Hits in Blackjack

In a 6 deck Blackjack game, what is the probability of a player hitting 4 times and still having a hand total of 20 or less, allowing for the option to draw a 7th card? What is formula to find this?
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0answers
15 views

how to related a weakly convergent random variable with its k-th moment

Let $\{X_n\}$ be independent random sequence with zero mean and unit variance. Suppose $$S_n:=\sum_{m=1}^n \frac{X_m}{\sqrt{n}} \Rightarrow X\sim \mathcal{N}(0,1)$$ holds. (Here "$\Rightarrow$" ...
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0answers
15 views

Simple bounding question for an expectation with truncating function

Let $\{X_m\}$ be independent random sequence. I want to show the following result Given $E[X_m^2]:=\sigma^2 < \infty$ and $$0 = \mathop {\sup }\limits_m P\left( {\left| {{X_m}} \right| > ...
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0answers
19 views

necessary conditions for these conditionals to be consistent with some joint distribution

Let $A$, $B$, and $C$ be random variables taking discrete values in the set $\{0,1\}$. I'm trying to find necessary conditions such that the conditional distributions $$X\mid Y,\,Y\mid Z,\,Z\mid X$$ ...
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1answer
35 views

Requesting deeper understanding of binomial coefficient

I noticed that $\binom {52} 4$ * $\binom {48} 1$ is $5$ times that of $\binom {52} 5$. So for example, if we were to draw $4$ cards from a standard deck then draw $1$ more, we cannot just say there ...
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2answers
15 views

Confidence Interval has no relation to the probability?

An Intro to Stats class has the following problem: Find and interpret the 90% confidence interval for the true mean The provided answer is this: ...
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0answers
9 views

How to show that a set of random strings has unit probability

I am encountering a problem where I want to show that the generation of a random string terminates in finite time with probability one, where the termination is condition is reaching an element of a ...
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0answers
35 views

Problem of points - probability [on hold]

Suppose a series of games is played between two teams and that in any individual game, one of the teams wins and the other team loses. Let the probability that team A wins an individual game be p and ...
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0answers
18 views

Probability of something happening in multiple trials

I have a similar question to this: How to calculate the probability of rolling 6 at least 5 times in a row, out of 50 tries? Unfortunately, while I understand basic maths and probability, the answer ...
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3answers
24 views

probability with percentage independence

There is 15% chance of choosing blue marble, 65% chance of choosing a purple marble, and 20% chance of choosing a black marble, there will be 18 total draws, what is the probability that you will ...
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0answers
15 views

Probability distribtuions [on hold]

A 10 metre by 10 metre plot of land is divided into 100 equally sized squares. Suppose that 300 seeds are randomly scattered on the plot of land. Use a suitable approximation to find the probability ...
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1answer
24 views

$X$ normally distributed in $\mathbb R^n$ iff components $x_i$ normally distributed?

We've had the normal distribution today in class and I was thinking about the following: Let $X$ be normally distributed, $X\sim N(a,\Sigma)$ with a symmetric positive definite matrix $\Sigma$ and ...
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2answers
22 views

Different approaches to N balls and m boxes problem

Suppose that you have N indistinguishable balls that are to be distributed in m boxes (the boxes are numbered from 1 to m). What is the probability of the i-th box being empty (where the i-th box is ...
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1answer
13 views

Expectation of a powered complex circular gaussian process

Assuming a complex circular zero-mean gaussian random process (or vector) $\textbf{x}$ $\left(\textbf{x}\sim \mathcal{CN}\left(0,\sigma^2\right)\right)$. $\mathbb{E}\{\textbf{x}\}=0$. The question ...
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0answers
18 views

Probability of hitting two balls for $d\geq 3$

The hitting probability for balls centered at origin is $P_{x}(T_{B_{r}(0)}<\infty)=\frac{r^{d-2}}{|x|^{d-2}}>0$ where $|x|>r$. So is it immediate that $$P_{x}[(T_{B_{r}(0)}<\infty) ...
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0answers
12 views

Joint CDF of Independent RVs [on hold]

Can anybody knows how to find joint CDF of $P(X_1^2<p_1,X_1Y_1<p_2,X_1Y_2<p_3,Y_1^2<p_4,Y_2^2<p_5) = ?$ where $X_i$ and $Y_i$ are independent zero mean Gaussian random variables. ...
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0answers
11 views

Function of Nakagami Distribution

Does anyone know what the distribution of the sum of squared Nakagami is? $$\sum_i^n X_i^2$$ $$X_i\sim \text{Complex Nakagami-m }$$ Is the distribution Erlang? Is the distribution the same as ...
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1answer
8 views

How to interpret this variance

If I have a probability measure defined by $P( \Omega ) = \int_{\Omega} (1-a) \delta(x) + a \delta(x-a^2) dx,$ then I noticed that the variance is given by $a^5(1-a)$. This is somewhat strange, cause ...
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3answers
37 views

Number of possibilities to draw from a card deck isn't an integer - where's my error?

I have a deck of 40 cards containing A,K,Q,J,10 of the 4 suits, each twice. I want to calculate the number of possibilities to draw 4 cards ignoring the suit. For the first card, I have 5 ...
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0answers
11 views

Decomposing factorized entropy

I am trying to figure out how the equation for factorized entropy below is derived. The equation for entropy is $H(Q) = -\sum_x Q(x)\log Q(x)$ where $Q$ is a probability distribution. Let $Q(x) = ...
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0answers
19 views

how to find expected value with toys? [duplicate]

A couple years ago, Burger King was giving a Dragon Ball Z toy in every kids meal. There were 6 unique toys that you could collect. Lets say I randomly selected a toy in each kids meal. What is the ...
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1answer
10 views

Zero hitting probability for positive measure sets in $\mathbb{R}^{d}$

In $d\geq 3$, we have that BM is transient a.s. i.e. $lim_{t\to \infty}|B_{t}|=\infty$. But does this imply $P_{x}(T_{A}<\infty)=0$ for some type of Borel sets $A\subset \mathbb{R}^{d}$ with ...
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0answers
14 views

What is the magnitude of Complex random variable Gaussian Case?

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$ $$X_2 \sim \mathcal{CN}(0,\sigma)$$ If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...
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0answers
8 views

Find function of error in sampling

I was given a teaser that I can only figure out half of. Imagine there is a dartboard that is centered at (0,0). Darts are thrown and the coordinates are modeled ...
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1answer
23 views

marginal pdf of a exponential distribution

Problem Let $X$ have the pdf $f(x)=e^{-x}$, $x>0$ and $Y$ have the pdf $f(y)=e^{-y}$, $y>0$. Assume that X and Y are independent Find the pdf of $U=X+Y$? Solution Since X and Y are indep. it ...
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1answer
13 views

When to use Binomial or Neg Binomial?

I have a problem that I'm not sure which distribution to use: 12 Toll employees were let go for taking more than 25,000 dollars in tolls. Lets say that one of the people let go on one day collected ...
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2answers
15 views

Can't see how implication on definition of Martingale was arrived at

A Martingale is a discrete time stochastic process $Z_1, Z_2, ..., Z_n$ for any time $n$ that satisfies $E[|Z_n|] < \infty$ $E[Z_{n+1}| Z_0, Z_1, ..., Z_n] = Z_n$ By the linearity of expectation ...
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0answers
25 views

Prove whether a particular function is concave

Given the following equation: $$V(w) = - \frac{\alpha}{2} \left[ y_1(w) + y_2(w) + \int _{-\infty}^{+\infty} \vert y_1(w) - y_2(w) - x\vert f_{T1}(x)dx\right] \\- \beta \int _{w - y_1(w)} ^{+\infty} ...
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0answers
25 views

Law of iterated logarithm proof

I am trying to master this proof of iterated logarithm. However, I get stuck at the last part. Here is a link In the last two line at fourth page. We calculate the probability that: $$ (*) ...
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0answers
68 views

An elementary annoyance

I'm going through some notes and I'm having problem understanding an inequality: The objects involved are: $X$ is a real-valued random variable with mean zero. We consider $n$ identical copies ...
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1answer
21 views

Birthday Problem for n number and x sharing it togetther?

There is an inter-college debate competition in which 40 students are participating. What is the probability that at least 4 of them have their birthdays in the same month? Now take the number is N ...
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0answers
23 views

Show conditional distributions are consistent with a joint distribution if ratios form a rank 1 matrix

Let $X$ and $Y$ be random variables taking discrete values in $\{1,\ldots,N\}$ and suppose the conditional distributions $P[X=i|Y=j]$ and $P[Y = j|X = i]$ are nonzero for all $i$ and $j$. I'm trying ...
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0answers
18 views

Question regarding double integrals

Regarding the Buffon's needle case for long needles of length $ l>t, $ (the distance between the parallel lines on the floor), we need to solve the integral $$ \int_{\theta=0}^{\frac{\pi}{2}} ...
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3answers
41 views

8 questions on true/false quiz. Expected number of correct answers with a given probability of correct answer of a question “i”.

The odds/probability of a correct answer of question $i$ is $p=1-2^{-i}$ What is the expected number of correct answers? My attempt: $$(1-2^{-1})\cdot (1-2^{-2})\cdot (1-2^{-3})\cdot ...
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2answers
37 views

How to show that the integral of bivariate normal density function is 1?

How to show the following? $$\large \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{2 \pi \sqrt{1-\rho^2}} e^{-\frac{x^2+y^2-2 \rho x y}{2(1-\rho^2)}} dx\ dy=1$$
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2answers
29 views

For a non-negative absolutely continuous random variable $X$, with distribution $F$. Why is $\lim_{t\rightarrow \infty}t(1-F(t))=0$?

So I am given a non-negative absolutely continuous random variable $X$ with distribution $F$, and density $p_X$. I am given the definition of expectation using simple functions and the survival ...
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3answers
34 views

Suppose you roll a pair of fair dice. What is the probability that the number of dots on the two dice sum to either $5$ or $10$?

Suppose you roll a pair of fair dice. What is the probability that the number of dots on the two dice sum to either $5$ or $10$? (a) $5/36$ (b) $7/36$ (c) $11/36$ (d) $4/36$ So here are the possible ...
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1answer
13 views

Probability of Normal Distribution

Let's say that 10 sumo wrestlers were to squeeze into an elevator that could only hold a max capacity of 2300 pounds. Let's say that the weight of the sumo wrestlers is normally distributed with a ...
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1answer
27 views

How to calculate $\mathbb{P}[Y\in F|X]_{\omega}$

Here I have an exercise of book: Probability and Measure of PATRICK BILLINGSLEY of conditional probability in the page 442, exercice 33.4 (b): Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability ...
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2answers
37 views

Prove $\mathsf E(N)=\sum_{i=1}^\infty \mathsf P(N\geqslant i)$

We want to prove that $$\mathsf E(N)=\sum_{i=1}^\infty \mathsf P(N\geqslant i)$$ We are given a hint that $$\sum_{i=1}^\infty\mathsf P(N\geqslant i)= \sum_{i=1}^\infty\sum_{k=i}^\infty \mathsf ...
2
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1answer
27 views

How to prove this convergence of a sum

How dos one prove the convergence of this sum $$e^{-m}\sum_{i=0}^\infty \frac{(mt)^i}{i!}=e^{m(t-1)}$$. I'm looking at the solution for a problem about probability generating functions and understand ...
3
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1answer
21 views

Is there a simplified form of this expression?

I have the following expression: $$ S = \sum_{k=1}^{K} \left(p_k \prod_{n=1}^{k-1}(1-p_n)\right) $$ All the $p_k$ are between 0 and 1. From numerical evaluations, I can see that when the $p_k$ are ...
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2answers
34 views

Why is the expected value of $|X|^p$ equal to $p\int_{0}^{\infty}y^{p-1}\mathbb{P}(|X|>y) dy$?

I'm trying to understand a passage from the book: A Basic Course in Probability Theory, Rabi Bhattacharya Edward C. Waymire, in the page 21. The calculation is the following: If $X$ is a random ...
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2answers
65 views

Why is this expectation finite?

For a random vector $X$, if $E (|c \cdot X|) < \infty$ for any vector $c$, then how can we show $ E(\|X \|) < \infty$? Thanks. Note: $\cdot$ means inner product.
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0answers
16 views

probability tennis games, player winning all games [on hold]

a tennis game has 12 total players of equal competence as in the same exact level, and each of these players plays each of the other players exactly once, Find the probability that one of the 12 ...