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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5
votes
1answer
44 views

Definition of conditional probabiliy as function dependent on $\sigma$-Algebra

I know that for events $A,B$ with $P(B) > 0$ the conditional probability is defined as $$ P(A | B) = \frac{P(A \cap B)}{P(B)}. $$ Of course by regarding $A$ as constant, and varying $B$ we get a ...
1
vote
1answer
46 views

About the distribution of balls in bins

Suppose we have $n$ balls and $n$ bins, and consider the following process: at stage $k$, we throw $\ln{n}$ balls into the bins, independently at random. We stop after $n/\ln{n}$ stages, when all ...
1
vote
1answer
22 views

Can a chain with repeated nodes still be considered a Markov chain?

The well-known Markov Property is that $$P(X_n = i | X_{n-1} = k_1, \dots, X_{n-j} = k_n ) = P(X_n = i | X_{n-1} = k_1) $$ Suppose we lay out some stochastic model in the following transition ...
2
votes
1answer
23 views

Determine the density of this problem

Let $X$ and $Y$ be independent random variables with a common density. You know this density has support only within the interval $[a, b]$ and that it is symmetric around $(a + b)/2$ (but you are not ...
2
votes
2answers
31 views

Proof that Corr(X,Y) equals zero for uniform discrete R.V.

i) X is a discrete uniform r.v. on the set $\{-1,0,1\}$. Let $Y=X^2$ . Prove that $Corr(X,Y)=0$. ii) X is a discrete uniform r.v. on the set $\{-1,0,1\}$. Let $Y=X^2$. Are X and Y independent? ...
0
votes
1answer
23 views

Prove this random variable has support in the first quadrant only

Let $f(t)$ be a density with mean $\mu$ and variance $\sigma^2$ with support on the positive half line $(t>0)$. Now show $$g(x,y) = \frac{f(x+y)}{x+y}$$ has support only in the first quadrant. ...
0
votes
2answers
39 views

PDF of Gamma R.V. [closed]

I know that $X \sim \exp(λ)$, $Z\sim \exp(λ)$ and $Y\sim \exp(λ)$ for $λ>0$. I also know that all three: $X, Y$ and $Z$ are independent. How do I find a pdf for $X+Z+Y$?
2
votes
1answer
21 views

Sum of uniformly distributed random variables in a given range

I am trying to find the sum of n uniformly distributed i.i.d random variables in the range [0-W]. I am aware that if the variables are distributed in the interval (0,1) then their convolution is given ...
3
votes
2answers
92 views

$4\times 4$ matrix game, covering $9$ of $16$ squares for even money bet. Good bet or not?

In a hypothetical game, person A offers a challenge to person B saying that A gets a $4\times4$ playing board ($2$ dimensional matrix) and gets to roll a special $16$ sided fair die such that each of ...
1
vote
1answer
47 views

Expected value of a Poisson variable conditioned on sum [duplicate]

Setting $$X_1 \overset{d}{\sim} \operatorname{Poisson}(\alpha_1)$$ $$X_2 \overset{d}{\sim} \operatorname{Poisson}(\alpha_2)$$ $$S = X_1 + X_2$$ Find $E[X_1 | S =n]$ My argument is that since $X_1 + ...
1
vote
2answers
14 views

Stats probability addition rule, multination rule

The directions are to calculate the following probability based on drawing cards without replacement from a standard deck of 52. What is the probability of drawing a 2 or a king on the first draw and ...
0
votes
1answer
57 views

Expected value of this deceptively simple variable

Setting: $X \overset{d}{\sim} \pmb{U}[-1,1]$ and $$\begin{align*}&Y = |X|\\[0.4cm]& Z = \begin{cases}\dfrac{X}{|X|}, & \text{ if } X \neq 0,\\[0.2cm] 0,&\text{ otherwise ...
1
vote
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
38 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 ...
1
vote
1answer
34 views

Compare expectations [closed]

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. I was asked this in an interview. No other information was ...
1
vote
3answers
50 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
31 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 ...
-4
votes
1answer
34 views

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

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
26 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
32 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
39 views

A problem on convergence. [on hold]

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
16 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
58 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?
-4
votes
1answer
44 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
56 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
51 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
22 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
19 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
23 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 $Y = 4X_1 – X_2$ [closed]

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 ...