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
37 views

Polynomial Chaos: How are the pdfs calculated from the response surface?

Lets assume one has the following response surface: $y(x,\xi) = \sum^N_{i=0} c_i H_i(\xi)$. Where $\xi$ is Gaussian and $H_i$ is the $i^{th}$ Hermite polynomial. I've seen a lot of papers show the PDF ...
2
votes
1answer
38 views

Find the unit vector so that this condition is true.

Let $(X_1,X_2)$ be jointly normal with density $$\phi(x_1,x_2;\rho) = \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left(\frac{-1}{2\sqrt{1-\rho^2}}(x_1^2 - 2\rho x_1x_2 + x_2^2)\right)$$ Find unit vector ...
0
votes
1answer
19 views

Random Variable probability summation tweaking

I can't seem to figure out what they do to get to the bottom
5
votes
1answer
89 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 ...
2
votes
1answer
148 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
58 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
27 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
37 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
37 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. ...
2
votes
1answer
78 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
135 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
136 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 + ...
2
votes
2answers
23 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
59 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
114 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 ...
1
vote
2answers
1k 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
129 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 ...
1
vote
2answers
560 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
44 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 ...
5
votes
1answer
137 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
58 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 ...
0
votes
0answers
96 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
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3answers
77 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
98 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 ...
0
votes
1answer
70 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???
2
votes
0answers
154 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
31 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
66 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 ...
1
vote
1answer
54 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
51 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 ...
-4
votes
1answer
69 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
1answer
70 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
41 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
1answer
79 views

question on uniformly distributed random variable

Let $X$ be a random variable uniformly distributed on the interval $[0,10]$ and zero elsewhere and let $Y$ be another random variable uniformly distributed on $[0, 20]$ and zero elsewhere. Assuming ...
1
vote
0answers
62 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 ...
0
votes
1answer
24 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, ... ...
1
vote
1answer
39 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 ...
2
votes
1answer
84 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
354 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
vote
1answer
49 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
1answer
31 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 ...
5
votes
1answer
79 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 ...
2
votes
3answers
118 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
1answer
76 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 ...
2
votes
1answer
554 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 ...
0
votes
1answer
306 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 ...
3
votes
3answers
63 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 $4$. ...
1
vote
1answer
58 views

Explain the result of this urn problem?

Suppose n balls are distributed in m urns. The probability that the first r urns receive k balls is $$\frac{\binom{n}{k}r^k(m-r)^{n-k}}{m^n}$$ I am most confused about the $r^k$ part. I know there ...
2
votes
1answer
35 views

distribution of the difference of discrete uniform RVs

Let $P_1, P_2$ be independent discrete uniform random variables on $\{0,1,...,k\}$. Suppose we want to compute $$\mathbb{P}(P_1 > P_2).$$ Is the best approach to see $\mathbb{P}(P_1 > P_2) = ...
2
votes
2answers
57 views

St.Petersburg Paradox and Bernoulli's quote

I was reading about St.Petersburg paradox, and understood the proof that $\frac{S_n}{n\log n} \overset{P}{\rightarrow}1$. The textbook then quotes Bernoulli: "There ought not to exist any even ...