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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1
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0answers
55 views

probability of 1s next to each other in sequence of numbers

I have a sequence of binary numbers (zeros and ones) and I'm trying to find the probability that $2$ ones will be next to each other. For example, I have something like: $11000$. So if I have two ...
0
votes
2answers
28 views

Probability coupon collection question - nth coupon is a new type?

I'm just solving some probability problems in preparation for my exam, and I stumbled upon this one which I cannot tackle: Suppose that you continually collect coupons and that there are $m$ ...
3
votes
1answer
31 views

Conditional probability for two normal distributed variables.

I haven't had to do much with probabilities since university, so please excuse if this is trivial or the question is not well specified. Let $X$ and $Y$ be two independent, normally distributed ...
0
votes
2answers
28 views

Defining median for discrete distribution

In probability theory, a median of a probability distribution is a number $M$ such that the CDF of this distribution $F_\xi(x)$ satisfies $F_\xi(M)=\frac{1}{2} \tag1$ This works for continuous ...
0
votes
2answers
37 views

Suppose $P(X \in B) \in \{0,1\}$ for all $B \in \mathcal B(\mathbb R)$. Show $X = c$, $P$-almost-surely.

Let $(\Omega, \mathcal F, \mathcal P)$ be a probability space and let $X$ be a random variable. Suppose $P(X \in B) \in \{0,1\}$ for all $B \in \mathcal B(\mathbb R)$. I want to show that there ...
4
votes
1answer
61 views

notation (ab)use for random variables, distributions, pdfs/pmfs

This question is about notation for random variables (RVs), distributions and pdfs/pmfs and their common (ab)use as I recently got confused. Let $X,Y$ denote random variables. First, notations I ...
3
votes
2answers
38 views

Almost surely convergence of the sequence

Let ${X_n}$ be a sequence of independent and identically distributed, square integrable random variables. Write $ u = E(X_n)$. Study the almost sure convergence, as $n \rightarrow \infty$, $$S_n ...
2
votes
0answers
31 views

Master equation of chemical reaction

I have about the construction of master equation for chemical reaction i.e. I have to construct differential equations for the probability mass function for the number of particles A, B and C. When ...
0
votes
1answer
26 views

Compare expectations [on hold]

X and Y are two random variables. How would you compare $E[XY]E[XY]$ with $E[X]E[XY^2]$ ? You need to tell which of these is greater/smaller.
1
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3answers
47 views

Probability that hard drive is defective

Suppose a manufacturer produces batches of 100 hard drives. In a given batch, there are 20 defective ones. Quality control selects two hard drives to test at random, without replacement, from the ...
0
votes
0answers
30 views

How to prove geometric mean is smaller than the arithmetic mean for a continuous distribution?

For discrete probability distribution, the geometric mean is defined as ${{\rm{E}}_{\rm{G}}}X = {\mu _G} = \sqrt[{\mathop \sum \limits_i {p_i}}]{{\mathop \prod \limits_i x_i^{{p_i}}}} = \mathop \prod ...
-5
votes
1answer
32 views

on an average $1$ vessel in every $10$ is wrecked [on hold]

If on an average $1$ vessel in every $10$ is wrecked. The probability that out of $5$ vessels expected at least $4$ will arrive safely is????
0
votes
1answer
23 views

Out of 3n consecutive positive integers…

Out of 3n consecutive positive integers, 3 are chosen at random without replacement. The probability that the sum of these numbers is divisible by 3 is???
-1
votes
1answer
38 views

Expected value of an exponential smaller than other exp [closed]

Lets say we have the following independent variables: $X\sim\exp(a), Y\sim\exp(b)$ and i want to find out the expected value of $X$ given that $X < Y$. that is $E[X\mid X < Y]$ How should i ...
2
votes
0answers
17 views

Why does Average Log Likelihood

The average log likelihood $$L(W,X) = \frac{1}{N}\sum_{1}^{N} log(p(x_n;W))$$ as defined by the authors in http://www.gatsby.ucl.ac.uk/aistats/fullpapers/217.pdf (first equation, first page, right ...
0
votes
2answers
26 views

How do I find (E|F')?

Assume ' is equal to not or complement here. Alright, you are given the following information: p(E)= 1/3 p(F)=1/2 p(E|F)=2/5 You are asked to find ...
0
votes
2answers
31 views

Probability for smallest and greatest

You have to deposit money five times. What is the probability that the first is the greatest and the last is the smallest ? ( five deposits are all different). Answer : 1/20 I did total number of ...
-2
votes
0answers
38 views

A problem on convergence.

Let $X_n$ be an iid sequence of non negative random variables (with $X_n<\infty$ almost surely) which have a common distribution with independent random variable $X$. How to prove the following: ...
1
vote
1answer
14 views

Understanding the proof of an Ergodic theorem for Markov chains

An ergodic theorem for Markov chains is as follows. If a Markov chain $(X_n)_{n \ge 0}$ is irreducible and has an invariant distribution $\pi$, then $$\frac{1}{n} \sum_{k=0}^{n-1} f(X_k) \to ...
1
vote
1answer
27 views

Conditional Expectation with Respect to “Y” as a Polynomial in “Y”?

I was reading on conditional expectation online when I came to this curious passage: I can easily understand that $\mathbb E[X|Y]$ can be seen as a function of $Y$: for any $\omega\in\Omega$ in the ...
0
votes
0answers
10 views

Is there a measure of probability gain that “normalizes” for diminishing returns?

Let's say we have a collection of of boxes that each have some known probability of containing an item I am looking for. I am given a few different answers on which order I should check them in, and I ...
-5
votes
4answers
55 views

How many possible 1mb files are there? [closed]

If you look at all combinations of data that can be stored in a 1mb file, how many are there before you have every possible 1mb file? How much space does that take up?
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votes
1answer
43 views

Almost sure convergence problem.

Let $(X_n)_{n\geq 1}$ be independent random variables: $X_n=n^2-1$ with probability $\frac{1}{n^2}$ and $X_n=-1$ with probability $1-\frac{1}{n^2}$. Let $S_n=\sum_{k=1}^{n}X_k$. How to prove that ...
1
vote
0answers
37 views

Phase trasition of $f(x)$ on random graph $G(n,p(n))$

Random graph $G(n,p(n))$ and graph $H$, which shown below, are given. I'm in need to find $f(x) : f(x) > 0$, such as: if $lim_{n \to \infty}p(n)f(n) = 0$, then asymptotically almost surely G ...
0
votes
3answers
31 views

Probability: Minimum Questions

My professor gave us this question in class in the last lecture saying he did this one year in his introductory classes. I don't even think this can be solved (well, with what we learned in class at ...
0
votes
0answers
34 views

Nullifying columns of a matrix by nullifying rows

Let $A$ be a real rectangular matrix. Each column of $A$ is a nonzero vector. Now each row of $A$ is nullified with probability $p$, all independently of each other. What is the probability that ...
0
votes
1answer
46 views

question on uniformly distributed random variable

Let $X$ be a uniformly distributed random variable on the interval $[0,10]$ and zero elsewhere and let $Y$ be another uniformly distributed random variable on $[0, 20]$ and zero elsewhere. Assuming ...
-7
votes
0answers
21 views

What is the chance I can ake the playoffs in one giving the following odds? [closed]

I am in three $10$ person leagues in which I have a $60\%$ chance in two leagues to get to the playoffs and a $40\%$ chance in the third one. What is my overall percentage chance that I will make the ...
1
vote
0answers
57 views

Really Big Decimals

Is there a probability that in a number such as e (2.7182818284590452353602874713527....) there will reach a point, no matter how long it takes (or how many blown-up processors), where mathematicians ...
-3
votes
0answers
24 views

help with random variables. [closed]

Let $X$ and $Y$ be random variables with a joint pdf $f_{x,y}(x,y)= C$ for $0 < X+Y < 1$, $0 < X < 1$ , $0 < Y < 1$ a. Find $C$ so that this is a valid joint pdf b. Find ...
1
vote
1answer
20 views

Is there a rule that can be used to easily approximate the pdf(x) for normal distribution?

Given the Normal Distribution with mean Mu and variance Sigma. With the respect to the rule of 3 Sigma, can one use similar estimations for the value of probability density function within 1, 2, ... ...
0
votes
1answer
16 views

inequality for real-valued Gaussian sums

I saw the following Lemma in an article: Let $\mathbf{b}\in \mathbb{R}^N$ be fixed, and let $\mathbf{\epsilon}\in \mathbb{R}^N$ be a random vector whose N entries are i.i.d. random variables drawn ...
-4
votes
0answers
29 views

Find expectation and variance [closed]

Let $X$ be a uniformly distributed random variable on the interval $0<x<10$ and zero elsewhere and let $Y$ be another uniformly distributed random variable on $0<y<20$ and zero ...
2
votes
1answer
38 views

Conditional Gambler's Ruin

I've learned about the most canonical gambler's ruin problems, but what if winning or losing on a previous turn changes the probability of winning or losing on the following turn? Say each turn I ...
3
votes
0answers
50 views

Best book for self-study on the foundations of probability

After some selection, I have three "candidates" books to purchase in order to study by myself the foundations of the theory of probability, at a level that I can define as "high undergraduate"/"low ...
-1
votes
1answer
22 views

Mean and Variance of probabilities [closed]

For a certain commodity which you buy, you can make either a $500$ profit with probability $0.5$ when you sell it, or $200$ with probability $0.3$ or lose $100$ with probability $0.2$. a. Find the ...
-3
votes
0answers
36 views

Great wisdom is need here…can YOU help? [closed]

a referendum is conducted with twenty five people given the chance to vote yes or no. each ballot box must contain at least 8 votes each how many possible outcomes are there? order of picks do not ...
-1
votes
1answer
27 views

Find the probability density function of $Y = 4X_1 – X_2$ [on hold]

Let $X_1$ and $X_2$ be independent normal random variables with means $23$ and $4$ and variances $3$ and $1$, respectively. Find the probability density function of $Y = 4X_1 – X_2$. No clue about ...
0
votes
0answers
15 views

Guessing the score [closed]

Every week in your local bookies they host a guessing game. They pick 11 random footballers from this weeks matches and ask everyone to rate their performance this week from 1 to 5 (whole numbers ...
1
vote
1answer
16 views

Probability of winning a game similar to bingo

I was trying to do the following question: I have attached the solutions and I am specifically confused about how they got the $${20 \choose 2}$$ the numerator of the first part. I usually post ...
1
vote
0answers
17 views

Robustness of Markov Chains

A Markov Chain on a measurable space $X$ is uniquely determined by a stochastic kernel $P$ on $X$. Let $\mathsf P_x$ denote the probability on paths generated by $P$ and the initial condition $x\in ...
0
votes
0answers
35 views

The efficiency property of estimator for second moment. [closed]

Please help to improve the efficiency property of estimation for second moment. Statistical population is normally distributed. Sorry for my bad english, if something is wrong. Thanks in advance.
4
votes
1answer
62 views

Why is this intuitive method valid?

Problem. There are $2$ white and $3$ black balls in the urn. A person randomly picked $2$ balls and put $1$ white ball. What is the probability of the event that the next randomly-picked ball would be ...
1
vote
3answers
50 views

An intuitive understanding of the equation $Var(X)=E(X^2)-E(X)^2$

I know the equation $Var(X)=E(X^2)-E(X)^2$ and its proof. After reading the textbook of mine, I found that this equation has been used in a lot of place. I want to know whether there is an intuitive ...
0
votes
0answers
27 views

Match this urn problem to a distribution

An urn initially contains r red balls and b black balls. A holding area outside the urn initially contain no balls. Balls are randomly chosen from the urn and: the chosen ball and the balls in the ...
0
votes
1answer
39 views

Distribution of transformed random variables

We have that f is a density w.r.t the lebesgue measure $m$ for a probability measure on $\mathbb{R}$, that f is continuous and strictly positive. X and Y are to random variables s.t. the distribution ...
1
vote
1answer
19 views

Tower Property and Variance of a Random Variable (Lightbulb problem)

Consider the following question: Type i light bulbs function for a random amount of time having mean (mew)i and standard deviation (sigma)i; where i = 1; 2. A light bulb randomly chosen from a bin of ...
1
vote
0answers
43 views

Estimating the mode

I am interested in the following problem and after searching I am surprised I can't find anything useful about it. Consider a multiset $A= \{a_1,\dots, a_n\}$ of integers. Say you sample elements of ...
0
votes
1answer
65 views

Linear transformation of random variables

We have to stochastic variables X and Y, and we define $ \begin{pmatrix} \tilde{X} \\ \tilde{Y} \end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X \\ Y ...
2
votes
3answers
43 views

Calculating the mean and variance of a distribution

Suppose $$P(x) = \frac{1}{\sqrt{2\pi\cdot 36}}e^{-\frac{1}{2}\cdot (\frac{x-2}{6})^2}$$ What is the mean of $X$? What is the standard deviation of $X$? Suppose $X$ has mean $4$ and variance $4$. ...