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
13 views

Inference on $P\left(\left.\sum_{i=1}^{N}X_{i}\right|\sum_{i=1}^{N}X_{i}^{2}\right)$ when $X_{i}\sim\mathcal{N}\left(0,1\right)$?

Let $$X_{i}\sim\mathcal{N}\left(0,1\right)$$ Hence: $$\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)$$ and $$\sum_{i=1}^{N}X_{i}^{2}\sim\chi^{2}\left(N\right)$$ What can be said about ...
1
vote
0answers
18 views

Transformation of probability density function

I'd like to compute the pdf of $w= g_1(x) = \frac{x}{1+e^{-x}}$ in dependence of the density $f_x(x)$. As I was not able to write the inverse function of $g_1(x)$, I tried the following approach: I ...
1
vote
1answer
15 views

Square with different densities. Computing probability.

I have a question about computing P(Y<0.5). Inside this square [-1,1] x [-1,1] we have different density function f(x,y). We can do it directly by counting area and it is 0.75. Because 4 is area ...
3
votes
2answers
61 views
+50

Poisson events distributed uniformly in a given time

It is given that $4$ Poisson events occur between $12:00$ to $13:00 $ (interval denoted by T). Intuitively, Why the probability of each event to occur at time $t ...
3
votes
2answers
42 views

Characteristic function with modulus 1 implies degenerate distribution

Let $X$ be a random variable with characteristic function $\phi(\ )$ satisfying $|\phi(t)|=1$ for all $|t|\leq 1/T$ with some $T>0$. Show that $X$ is degenerate, i.e., there is $c$ such that ...
0
votes
1answer
535 views

For each of the following, determine the constant c so that f(x) satisfies the conditions for being a p.m.f

For each of the following, determine the constant c so that f(x) satisfies the conditions for being a p.m.f. for a random variable X. c) f(x) = x/c, x = 1,2,...,n d) f(x) = c/(x+1)(x+2), x = ...
8
votes
2answers
2k views

Given two randomly chosen natural numbers, what is the probability that the second is greater than the first?

Suppose that there exists some apparatus that, when prompted, displays a random natural number (i.e. it picks an integer uniformly from the range $[1, \infty)$). A man writes down a number generated ...
1
vote
0answers
33 views
+100

Is $F_n\to F_{\infty}$ equivalent to $\lim_{n\to\infty}\int\phi dF_n=\int\phi dF_{\infty}$ for every $\phi \in C(R)$?

If $F_1,F_2,...,F_{\infty}$ are distribution functions. Is $F_n\to F_{\infty}$ equivalent to $\lim_{n\to\infty}\int\phi dF_n=\int\phi dF_{\infty}$ for every $\phi \in C(R)$? I intuitively think this ...
1
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0answers
16 views

Likelihood at least 2 out of n numbers are visible to each other in Z^n

Two points in $ \mathbb{Z}^n $ are said to be visible to each other, if they can be connected by a straight line, which doesn't intersect any points of $ \mathbb{Z}^n $ In Apostol's book "An ...
-1
votes
1answer
27 views

What's the summary probability of an event if it increases over time?

I'm having trouble calculating this one. Say there are two steps an event occurs with certain probability: 60% 70% What is the probability that an event occurs by the time second step is reached? ...
0
votes
0answers
19 views

Markov Property Definition

Let $(X_t)$ be a stochastic process on $(\Omega, \mathcal F, \{\mathcal F_t\}, \mathbb P)$. The typical definition of the Markov property is $\mathbf{P}(X_{t+s} \le x \, |\, \mathcal F_t) = ...
0
votes
0answers
12 views

Is correct my Procedure about Joint Distribution for independent random variables

$ y_i, i=1,2...n$ are random variables are linearly independent For $y_i \sim Ber(p)$ $(p^{x_1}q^{1-x_1})(p^{x_2}q^{1-x_2})\bullet \bullet \bullet (p^{x_n}q^{1-x_n})$ ...
-1
votes
0answers
15 views

Collision detection for two moving objects

There are 2 objects $A$ and $B$. Both have 2 sensors. The sensors can measure a distance to another sensor. Let's say the sensors are $AF$ (front sensor for $A$), $AR$ (rear sensor for $A$), $BF$, ...
-2
votes
0answers
28 views

Counting math problems

1) Ann, Bobby, and Cece are randomly placed in a line with 26 people total. What is the probability that Ann is to the left of Bobby, and Bobby is to the left of Cece? Express your answer as a common ...
-3
votes
1answer
40 views

What are the expectations of $1/X$ and $1/(1-X)$ if x has a Dirichlet distribution? [on hold]

What are the expectations of $X^{-1}$ and $(1-X)^{-1}$ if $X$ has a Dirichlet distribution?
1
vote
1answer
38 views

Show $X$ and $Y$ are independent if we assume that $E[XY] = E[X] E[Y] $

Assume that $$E[XY] = E[X]E[Y]$$ Let $X$ and $Y$ be random variables taking two different values $a,b \in \mathbb{R}$. Show that X and Y are independent. Note: I've spent a long time on this ...
-2
votes
1answer
49 views

Conditional distribution of mixed process

$$ N(t)=(1-B).N_0(t)+B.N_1(t), \quad \quad \text{where B is Bernoulli($p$), $N_0(t) \sim \operatorname{Poiss}(\lambda_0 t)$ and $N_1(t) \sim \operatorname{Poiss}(\lambda_1 t)$}. $$ I suspect that ...
0
votes
1answer
15 views

Show that f is a density and find the corresponding cdf

$f(x) = \frac{(1+\alpha x)}{2} $ for $-1 \leq x \leq 1$ and $f(x) = 0$ otherwise, where $-1\leq \alpha \leq 1$. Show that $f$ is a density and find the cdf. I am mainly having trouble with finding ...
0
votes
1answer
7 views

Walking through the reduction of a cumulative probability function to a polynomial

Setup Define $P(p)$ as follows: $$ P(p) = \sum_{N_1-\phi \cdot N_2 \geq \theta} {n_1 \choose N_1} {n_2 \choose N_2} p^{N_1 + N_2}q^{n_1 + n_2 - N_1 - N_2}. $$ Here, $$ q = 1 - p. $$ The sum is ...
0
votes
2answers
29 views

Let $N$~Pois$(\lambda)$, $X|(N=n)$~Bin$(N,p)$, $Y=N-X$. Show $X$, $Y$ are independent and Poisson with parameters $\lambda p$ and $\lambda (1-p)$.

Any direction on this problem would be much appreciated. So far I know the joint distribution of $X$ and $Y$ is $\begin{align} \mathsf P(X=x, Y=y) & = \mathsf P(X=x, N-X=y) \\ & = \mathsf ...
-3
votes
1answer
32 views

Cats and Dogs = Idenpedent events

I did not get this question. Could you explain it to me? In a building for 24 apartments. It is known that there is only one dog in 8 apartments and a single cat in 6 apartments. How many apartments ...
-4
votes
0answers
17 views

Probabilityorchance [on hold]

i have a list of 9 people 1 of whom will be elected by a group of 65 people. The rules are this. Each person votes for three people. Each ballot must contain 3 different names. Each person complies ...
0
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0answers
13 views

Questions about solution to finding solution to mode of a binomial distribution

So i read over the solution presented by Andre Nicolas: finding mode in Binomial distribution But i have a few questions about the whole thing: 1) why did he set the ratio as $\frac{a_{k+1}}{a_k}$? ...
0
votes
1answer
44 views

Finding patterns in seemingly arbitrary pairs of numbers

I don't work (directly) in mathematics (I'm a programmer), but I see numbers every day. Today I came across an issue where some totals were off, and was sent a list of the last 9 examples of the ...
-1
votes
0answers
15 views

Probability Data Management [on hold]

A bag contains 54 black marbles and 63 white marbles. Use Pascal’s Triangle to determine how many combinations and how many permutations are possible if 7 marbles are drawn out of the bag.
1
vote
2answers
300 views

Random directions on hemisphere oriented by an arbitrary vector

Hy, i'm writing a raytracer, and for that I need to generate n random vectors that are inside an hemisphere oriented by the surface normal. Ideally, I would also like being able to restrict the rays ...
0
votes
1answer
19 views

Why distribution of multiple recursive random number generators is uniform?

I was reading the article of L'Ecuyer on random number generation. The title of this article is "Uniform Random Number Generation". One of the proposed PRNGs there, is multiple recursive random ...
2
votes
3answers
63 views

How do I find the constant C?

Consider a random experiment with a sample space $$S=\{1,2,3,⋯\}$$. Suppose that we know: $$P(k) = P({k}) = \frac {c}{3^k}$$ for $k=1,2,⋯,$ where c is a constant. Find c. Find $P(\{2,4,6\})$. Find ...
-1
votes
3answers
384 views

Where can i download “What are the chances?:Probability made clear” lectures videos? [on hold]

I want to learn probability through lectures videos and i would like to know where i can download The Teaching Company videos for the course titled "What are the chances ? :Probability made clear " I ...
-1
votes
2answers
42 views

PDF of $Y=\min(0,X)$ when PDF of $X$ is $\frac34(1-x^2)$ on $(-1,1)$

Let $X$ be a random variable with density $f(x) = (3/4) (1-x^2).$ Range is $-1 < x < 1.$ I have to find probability distribution of $Y = \min(0,X).$ I know that distribution function could be ...
2
votes
3answers
37 views

Inductively defined random variables

Let $X_0=1$, define $X_n$ inductively by declaring that $X_{n+1}$ is uniformly distributed over $(0,X_n)$. Now I can't understand how does $X_{n}$ gets defined. If someone would just derive the ...
0
votes
3answers
29 views

Uniform PDF for continuous variable, why does the probability values increase to 1, when its normalized?

Consider a "spinner": an object like an unmagnetized compass needle that can pivots freely around an axis, and is stable pointing in any direction. You give it a spin and see where it comes to rest, ...
3
votes
1answer
26 views

Connecting noodles probability question

I don't know how to solve this. You have 100 noodles in your soup bowl. Being blindfolded, you are told to take two ends of some noodles (each end of any noodle has the same probability of being ...
4
votes
0answers
23 views

Given two sets of $100$ samples of $10$ items from a $1000$ item set, what is probability that the two sets have non-empty intersection

Suppose two people go grocery shopping $100$ times each. Each time, they pick $10$ items randomly from the $1000$ items at the store. As a result, each person has $100$ randomly chosen baskets of ...
0
votes
1answer
40 views

What will be Terms after repeating this step(Differentiation and multiplication) F times.

I was solving a probability problem and got stuck on the following situation, where each x_i is independent of others: $$f=(x_1+x_2+..x_k)^N$$ I'm interested in the expression obtained after ...
2
votes
0answers
68 views
+250

The probability that two matrix vector products are equal

Consider a random $n$ by $n$ circulant matrix $M$ whose first row entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first ...
-4
votes
0answers
20 views

What is the Probability of drawing number less than 3 when a die os rolled? [on hold]

What is the Probability of getting number less than 3 when a die is rolled?
1
vote
1answer
17 views

Calculating the probability of a set of numbers appearing in a randomly-generated 3x3 grid

A challenge within a PC game I play features a 3x3 grid which contains all numbers from $1$ to $9$ in a random order. For example, this may be what a randomly-generated 3x3 grid looks like: ...
0
votes
1answer
354 views

Characteristic function and probability density function: Fourier or Inverse Fourier?

I have come across two contradicting definitions of characteristics function (CHF). In wikipedia CHF is defined as the inverse Fourier transform (FT) of probability density function (PDF) and at some ...
0
votes
0answers
19 views

Central Limits Theorem [on hold]

An instructor has 50 exams that will be graded in sequence. The times required to grade the 50 exams are independent, with a common distribution that has mean 20 minutes and standard deviation 4 ...
3
votes
3answers
32 views

Combinatorics question on group of people making separate groups

If there are $9$ people, and $2$ groups get formed, one with $3$ people and one with $6$ people (at random), what is the probability that $2$ people, John and James, will end up in the same group? ...
4
votes
1answer
115 views

Soft question: what are some elementary motivations of using functional analysis to study probability theory?

Recently I've become curious about the links between functional analysis and probability theory. What are some simple reasons why a functional analytic approach is preferable to a measure-theoretic ...
0
votes
2answers
46 views

'Probability method' - to what extent is it an actual proof?

Consider this: $\frac { C^{2n}_n } {2^{2n}} = \mathbb P (A) $ where A= { equal number of heads and tails in $2n$ throws of a fair coin } therefore the following assertion is true: $\forall n \ge ...
0
votes
1answer
346 views

finding unconditional distribution by integrating conditional distribution

Given $$ f_Y (y)= \begin{cases}\frac{1}{120} e^{-\frac{1}{120}y} &, y\ge 0 \\ 0, &, y< 0 \end{cases}$$ and $$f_{X|Y} (x|y) = \begin{cases}\frac{1}{y} &, x\in [0, y] \\0 &, ...
0
votes
1answer
17 views

Query about non-singular transformation of vectors

Suppose we are given a probability function, P (x^T (Y-z)≥0) , where ‘x’ is a vector, ‘Y’ is a random variable and ‘z’ is a known value. Now, suppose, we make a non-singular transformation w=Ax, ...
2
votes
2answers
33 views

What are the odds of spinning matching items in a slot machine?

Lets say we have a slot machine with $5$ reels. Each reel has $5$ different items on it. What are the odds of spinning $2, 3, 4$ and $5$ matching items? As I understand the probability of rolling a ...
2
votes
0answers
50 views

Probability of an Indisputable winner at Texas Holdem

What are the fraction of hands that can be classified as "indisputable winners" after the river is revealed in Texas Holdem? By "hand" I mean the 2 hole cards you have that no one else can see plus ...
0
votes
2answers
28 views

5-Card Poker Two-Pair Probability Calculation

Question: What is the probability that 5 cards dealt from a deck of 52 (without replacement) contain exactly two distinct pairs (meaning no full house)? Solution: ...
0
votes
0answers
35 views

Bookmaker's odds

Suppose a match can be completed in three ways: win, loose or draw. A bookmaker provide the following coefficients (including spread) for each case respectivelly: $c_1, c_2, c_3$. That is, if a player ...
0
votes
0answers
43 views

Additional Problems about conditional expectations

When I was a student, I was able to solve these problems, but now I can't because I forgot some important things. If you show me the solutions to at least two of them, I am sure, I will refresh it all ...