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 ...

learn more… | top users | synonyms (2)

0
votes
1answer
15 views

Probability question - picking balls from a bag one at a time

Suppose I have an unknown number of balls (N), each of a different color hidden in a bag. How many times must I draw a single ball, make a note the color and return it to the bag in order to be sure ...
0
votes
0answers
11 views

How does one define the Fourier transform of a probability distribution?

Say $p_X$ and $p_Y$ are two probability distributions on a $m$ element set. Then I see an equality written as, $$\sqrt{m} \vert \vert p_X - p_Y \vert \vert _2 = \sqrt{ \sum_{k=0}^{m-1} \vert ...
1
vote
0answers
17 views

Given Gaussian distributions of probabilities of a random variable, is it accurate to take every data point into account?

I understand the phrasing of the question above is a little vague, but I couldn't figure out how to phrase it more accurately, so please feel free to fix it if you have a better way after reading ...
2
votes
2answers
25 views

Are there two different notions of “conditional probability”?

This question comes from reading the discussion here. (1) If one is given a "probability measure" $P : F \rightarrow [0,1]$ mapping a Borel $\sigma$-algebra $F$ to $[0,1]$ then for two ``random ...
1
vote
1answer
22 views

probability to finding an answer in a book

An answer to a question can be in 5 books, the probability that the answer is in a book at library 1 is $\frac{1}{3}$, else the answer is in the left 4 book in library 2 (same likelihood). ...
0
votes
0answers
20 views

How do we obtain this inequality? A question concerning an argument in Stroock and Varadhan 1971

The problem comes from the article of Stroock and Varadhan [diffusion processes with boundary conditions (1971) ]. So far I have followed, but in the next page I got lost: I don't follow the ...
0
votes
1answer
8 views

Equivalence of the second moment of two random variables when their first moments and covariance with a third random variable are equal

I'm trying to check under which conditions the standard deviation of two random variables is identical when I know some properties about other moments of these random variables. I suppose that their ...
0
votes
0answers
7 views

Probability that one chi-squared random variable is less than other chi-squared random variable

I have two random variable $X=\mathcal N(\mu,\sigma^2)$ and $Y=\mathcal N(0,\sigma^2) $ independent to each other. Now, $Z=X^2$ and $W=Y^2$, are chi-square random variable with first degree of ...
0
votes
2answers
26 views

Probability That People Live On Different Floors.

There is a building with 3 floors, on each floor there are 4 apartments, and in each apartment lives 1 person. GIven that 3 people meet in the entrance of the building, what is the probability ...
1
vote
0answers
12 views

Probability number is divisible by half the square of a prime?

Let $p$ be a prime. What is the probability that a number of the form $\left \lceil \frac{p^2}{2} \right \rceil$ divides a random positive integer $n$. I have a solution that involves the Riemann-Zeta ...
0
votes
1answer
16 views

Equivalent definitions for expected value of random variable

Let $(\Omega, P)$ be a probability space. One definition for the expected value of a random variable $X$ is $$E(X)=\sum_{x\in \mathbb{R}} xP(X=x).$$ The notes I am reading say that this definition ...
-1
votes
0answers
20 views

Line at the cinema - probability theory

At the ticket booth at the cinema is are people queued up. The first person in the line whose birthday is the same as the person behind or in front of him, gets a free ticket. a) Carl is fifth in the ...
0
votes
0answers
23 views

Stirling aproximation [duplicate]

I was reading my book of stochastic processes, when suddenly appear the following approach $$n!\sim n^{n+\frac{1}{2}}e^{-n}\sqrt{2\pi}$$ From where this result comes? I looked on Wikipedia but ...
2
votes
0answers
26 views

Sources for simple probability brain teasers

I am searching for a book that can supply me with probability brain teasers, that can be solved using little arithmetic/mental math, paired with somewhat detailed solutions. Any suggestions? ...
0
votes
2answers
30 views

A real statistic for pay-per-click advertisement

After each click happens a "conversion" with probability $p$. All conversions are independent. Suppose there were $n$ clicks and $m$ conversions. What is the probability that $p>p_0$ for a given ...
0
votes
1answer
13 views

Finding probabilities from probabilty generating function

Given that I have a probability generating function for $Q$ given by $\dfrac{4s^{2}}{9-3s-2s^{2}}$, I want to find $P(Q = n)$ for $n \geq 2$. I understand that I could actually use the definition of ...
1
vote
2answers
38 views

How To Approach Dice Rolls

When asked about 2 dice roll, we do we count the result that both dice have the same number just one time and not two? If it is because we can distinguish between the two, so if the dice was colored ...
0
votes
0answers
14 views

Inference from $\operatorname{Cov}(X^2,Y)>0$ to $\operatorname{Cov}(X,Y)$

Let $X$ and $Y$ be non-negative random variables which take integral values, and such that $\operatorname{Cov}(X^2,Y)>0$. What could be said on $\operatorname{Cov}(X,Y)$?
2
votes
0answers
24 views

An interesting observation about Poisson random variables.

The pmf of a Poisson distributed random variable is given by \begin{equation} P(X=k) = \exp(-\lambda) \frac{\lambda^k}{k!}\,, \end{equation} where $\lambda>0$. The corresponding MGF is given by ...
1
vote
1answer
44 views

Proof that random variable is almost surely constant

If a random variable $X : \Omega \to \mathbb R$ is $\{ \emptyset, \Omega \}$-measurable, then it is constant. I want to generalise this result: Now if $\mathcal G$ is a $\sigma$-algebra such that ...
2
votes
0answers
22 views

Expected value (mean) of function from polyline

Suppose we have a polyline that has such properties: It consists of n segments First segment's ends are (0, 0) and ...
1
vote
0answers
50 views

A question of odds

Consider an experiment with four possible outcomes, and suppose that the quoted odds for the first three of these outcomes are as follows. What must be the odds against outcome 4 if ...
0
votes
2answers
39 views

Calculating a characteristic function two different ways gives contradictory results. Why?

I am trying to calculate a characteristic function directly and via the conditional distributions. I get contradictory results: Let $X$ and $Y$ be random variables defined on the same probability ...
0
votes
1answer
24 views

Probability for rolling an odd number and tossing a coin on heads

A coin is tossed and a die rolled. Find the probability of getting a head and an odd number. The answer is $\frac{1}{4}$. My reasoning is that rolling an odd number is $\frac{1}{2}$, and tossing a ...
0
votes
2answers
20 views

how do they calculate these following columns

I have these data: I am sorry the data is in Portuguese, and it is an image so I can't convert it to a table but the translate "probably" ( i am not a native speakers for Portuguese language) is: ...
0
votes
1answer
10 views

Cumulant-Legendre

I have a short question: So suppose $b=\text{ess sup} X<\infty$, where $X$ is a random variable on $\mathbb{R}$. Now take $\Lambda (u)=\ln \mathbb{E}[e^{uX}]$, the cumulant, and ...
2
votes
4answers
78 views

A combinatorial proof for $\binom mk$+$\binom m{k-1}$=$\binom {m+1}k$

I do realize that there is a elementary proof of this result which follows from applying the formula $$\binom mk=\frac{m \cdot (m-1) \cdot \ldots \cdot (m-k+1)}{k!}.$$ I do wonder if there is an ...
1
vote
1answer
23 views

Conditional probability of a zero inner product

Consider a random $n$ by $n$ matrix $M$ chosen uniformly over all $n$ by $n$ $(0,1)$ matrices and a random vector $v \in \{-1,0,1\}^n$ chosen uniformly as well. Let $X = Mv$. What is $$P(X_i = 0 ...
1
vote
0answers
18 views

Can anyone shed some light on the random variable which has the following characteristic function?

I have a random variable whose characteristic function is of the form \begin{equation} \mathbb{E}[e^{itX}] = \frac{(1-it)^a}{(1-2it)^{\frac{a}{2}}}\,, \end{equation} where $0<a<1$ I am not ...
0
votes
0answers
18 views

Continuous mapping theorem with density convergence

Let us consider a bivariate random variable $X\in \mathbb{R}^2$ with $pdf$ $f$. Also let, based on a sample of size $n$, let the the estimator of the density be $f_n(x)$ at $x\in \mathbb{R^2}$ and we ...
0
votes
1answer
15 views

Let X and Z form a random sample from a poisson dist.If Y=min( X,Z), what is P(Y=1)??

Let X and Z form a random sample of poisson distribution and define Y=min( X and Z) What is P(Y=1)?? I think Y is minimum of two. If X=1, then Z can be any number except 0 If Z=1, then X can be ...
1
vote
1answer
22 views

Comparing Percentiles of 2 Samples Drawn from the Same Distribution

Suppose I have two sets of numbers: $A=\{a_1,a_2,...a_{N_1}\}$ and $B=\{b_1,b_2,...b_{N_2}\}$ with $N_1<N_2$. WLOG assume that $a_i<a_j$ for all $i<j$ and similarly for $b_i$ and $b_j$. ...
0
votes
2answers
35 views

Dice Probability (increasing numbers)

If I have 6 regular dice, (each numbered 1-6): What is the probability that when rolled that each will be a different number.(each individual di is a different number from 1-6, but a random order) ...
0
votes
1answer
32 views

The probability of a number appearing in an approximation of an irrational number?

I was wondering if for the number Pi some numbers are more likely to appear than others, for example 3.141594 ... The number 1 appears twice does that mean that the probability for the number 1 ...
0
votes
1answer
18 views

How to calculate covariance of X and Y given joint probability

$X$ and $Y$ are dependent variables both normally distributed as $N(\mu-const, \sigma^2)$. I don't know what the joint distribution is. I know that when $const = 0$, then the joint probability ...
4
votes
2answers
43 views

Determining probability generating function for event “$SS$”

Given a sequence of Bernouilli trials, we have $P(S) = \frac{2}{3}$ with $0<p<1$. The event "SS" occurs on the $i$-th trial if we observe an $S$ on the $i$-th trial following a $S$ on the ...
1
vote
2answers
36 views

Combining Markov chains

If the following Markov chain relations hold: $$X \rightarrow Y \rightarrow Z,$$ $$Z \rightarrow W \rightarrow Y,$$ can we combine them to have $$X \rightarrow Y \rightarrow Z \rightarrow W ...
0
votes
1answer
22 views

Given probability of two elements being same in a list, find total number of unique elements

I have a list L, of numbers ordered randomly. Every number in the list is from a domain of $1$ to $100$ with the possibility of duplicates. If I point to(without removing) two numbers from the list ...
1
vote
3answers
40 views

Normal distribution exercise!

If a technician does not encounters any hardware problems, the time he requires to assemble a computer follows a normal distribution with a mean of $30$ minutes and a standard deviation of $3$ ...
1
vote
1answer
21 views

Given Nd6, what is the probability that the two highest are minimum 4?

So, my statistics knowledge is rather poor, so I would welcome a formula explanation to the question: given Nd6 (6-sided dice) what is the probability that the two highest numbers are at least a 4? ...
7
votes
2answers
62 views

Show rigorously that Pólya urn describes a martingale

We work with the famous Pólya urn problem. At the beginning one has $r$ red balls and $b$ blue ball in the urn. After each draw we add $t$ balls of the same color in the urn. $(X_n)_{n \in \mathbb ...
-5
votes
0answers
53 views

How many from 0 to 99999 [on hold]

How many times does the number 92789 appear in any sequence from 0 - 99999. If you know can you please include the formula.
0
votes
1answer
46 views

Probability of a rolling a dice $n$ times with $k$ faces

I need help calculating the probability of rolling $n$ dice with $k$ faces. So you have multiple dice, all with $k$ faces (number of sides on a dice) and you want to calculate the probability of a ...
1
vote
3answers
35 views

About discrete probabilities (Expected values)

Is my solution correct? Suppose two player (A and B) each one with 200,00 dollars toss a coin not balanced in a such way that the probability of head is $p$. Suppose yet that if the result obtained ...
0
votes
0answers
18 views

Find the constants for the independence of a random variable

The following is my question: Let $W(t)$ be a standard Brownian motion, $\xi\sim N(\mu,\sigma^2)$, and $\xi$ is independent of $W(t)$ for all $t\geq0$. Define $X(t)=t\xi+\lambda W(t)$, for some ...
0
votes
1answer
23 views

Markov Chains, reccurent and transient

Let the Markov Chain consisting of the states $0,1,2,3$ have the transition probability matrix ...
1
vote
4answers
57 views

Probability question involving infinite number of vertical chords in a 1 inch circle. [on hold]

Infinite number of vertical chords drawn on a circle with a 1 inch radius. What is the probability that a randomly picked chord is shorter than the radius? The answer should be $1 - .5√ 3$ or ...
2
votes
0answers
8 views

Likelihood that two markov chains are derived from the same transition matrix

Forgive me for my weak statistic background, hopefully what I'm asking makes sense. So some quick background, I have one markov chain from a data set and many additional chains that I'm producing from ...
2
votes
2answers
42 views

Does the distribution of a process on $\mathbb{R}^{[0,\infty)}$ uniquely define it?

Question: Can I have two different stochastic processes $(A_t)_{t \in [0, \infty)}$, $(B_t)_{t \in [0, \infty)}$ having the same distribution on $\mathbb{R}^{[0, \infty)}$ differ in some ways? ...
0
votes
1answer
44 views

An isosceles right triangle has legs of length 10. A pin is dropped into it and lands somewhere in the triangle where all places are equally likely.

What is the probability that it does not land within 2 units of any of the sides? From my calculations, I get that the smaller triangle has side lengths of 4,4, 4 root 2 (-2 at the right angle and ...