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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3answers
25 views
Question about Switching Between Random Variables
Find the density function of $Y = aX$, where $a > 0$, in terms of the density function of $X$.
Show that the continuous random variables $X$ and $-X$
have the same distribution function if and ...
2
votes
1answer
7 views
Normalizing a continuous distribution
I work at a help/tutoring center at my university. Today a kid came in with this problem. I've only studied math and haven't drifted into physics, but he had this problem:
Let ...
-1
votes
1answer
28 views
Probabilty of A and D but not B and not C
I am trying to work a probability question and I am stumped.
Imagine there are 4 signs on the side of the road and each sign has a "chance of being seen" equal to the following
sign A: .75
sign B: ...
0
votes
0answers
16 views
What is the average weight of a minimal spanning tree of $n$ randomly selected points in the unit cube?
Suppose we pick $n$ random points in the unit cube in $\mathbb{R}_3$, $p_1=\left(x_1,y_1,z_1\right),$ $p_2=\left(x_2,y_2,z_2\right),$ etc. (So, $x_i,y_i,z_i$ are $3n$ uniformly distributed random ...
4
votes
1answer
23 views
In a random graph of $n$ vertices, what is the expected value of the number of simple paths?
I am very new to discrete probabilty and was asked this question:
In a random graph $G$ on $n$ vertices (any edge can be in the graph with probabilty of $\frac{1}{2}$,) what is the expected value ...
1
vote
2answers
30 views
Probability inequality for sum of two random variables
Does anyone know if the following inequalities are right? If yes, what is the reference for them?
For random variable $x$ and $y$:
\begin{equation}
\mathbb{P}(x+y \ge x_0 + y_0) \le \mathbb{P}(\{x\ge ...
1
vote
1answer
45 views
+50
Converting Expected value to integrals and differentiating
Can you suggest me how to convert the following expected value function in to an integral and differentiate it with respect to $a$.
\begin{equation*}
g \equiv E \left[ \max \left( a + x-b,0 \right) ...
0
votes
1answer
33 views
Choosing random number $[1,n]$. What is the expected value of $f(x) = x^2$?
We have just started learning discrete probability and this question came up:
We choose a random number from $[1,n]$, and we let be $f(x) = x^2$ and $g(x) = 2^{-x}$.
I) What is the expected ...
1
vote
1answer
27 views
Bottom to top explanation of the Mahanalobis distance?
I'm studying Pattern recognition and statistics and almost every book I open on the subject I bump into the concept of Mahanalobis distance. The books give sort of intuitive explanations, but still ...
1
vote
2answers
27 views
Random Variables from [0,1] - Integration Limits
I was wondering if someone could help me understand the first steps I should take for solving the next problem:
Let $U$, $V$ be random numbers chosen independently from the interval $[0, 1]$ with ...
3
votes
3answers
85 views
Central Limit Theorem Definition
My friend and I have a bet going about the definition of the Central Limit Theorem.
If we define an example as a number drawn at random from some probability density function where the function has a ...
2
votes
4answers
72 views
Why is the Expected Value different from the number needed for 50% chance of success?
An event with probability $p$ of being success is executed $\frac{1}{p}$ times. For example, if $p=5\%$, the event would then be executed $20$ times.
The Expected Value for the total number of trials ...
0
votes
1answer
28 views
the distribution of the inverse of a standardized uniform variable
If $u$ is a standardized uniform variable, what is the mean and variance of $x=\frac{1}{u}$? What can be said about the distribution of $x$?
2
votes
6answers
125 views
Probability problem
I have $3$ coins, $1$ coin has $2$ heads (HH), 1 coin has $2$ tails (TT), $1$ coin has $1$ head and $1$ tail (HT). I toss the coin, it fells on my hand, and the side i see is a tail. What's the chance ...
3
votes
1answer
46 views
How to show $nP\{|X|>n\}\to 0$ as $n\to\infty$, but $X$ is not integrable.
How can I construct a random variable $X$ such that: $nP\{|X|>n\}\to 0$ as $n\to\infty$, but $X$ is not integrable.
0
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3answers
51 views
Distance between two Random Variables by comparing Cumulative Distribution Functions
Suppose $X$ and $Y$ are two random variables. Define the distance between $X$ and $Y$, $d(X, Y)$ as: $$d(X, Y) = \int_{-\infty}^{\infty}\left|\mathbb{P}(X < t) - \mathbb{P}(Y < t)\right|dt$$
...
0
votes
1answer
30 views
Is the Uniform family of Distributions dominated by the Lebesgue Measure?
The answer to this question should be fairly easy, but I can not just see it.
I want to say something like: let us consider a measure $P_{\theta}\in\mathcal{P}$ where ...
1
vote
1answer
32 views
Finding the expectation and characteristic function of a mixed distribution.
I'm having difficulty with a practice exam question. Here's a modified version.
First, some notation. Let $E$ denote the exponential distribution, and $B$ the Bernoulli distribution. Also, given a ...
0
votes
1answer
29 views
How does this expression arise: $\pi(10.5) = \phi (-z_{1-\alpha} + \sqrt{n} \frac{\mu_0-\mu}{2})$?
$X_i$ is $N(\mu,\sigma^2)$ distributed and the following is given $H_0: \mu \geq 12, H_a: \mu < 12$, and $\alpha=0.01$. I'm asked to calculate $\beta=P[TII]$ if in fact $\mu=10.5$
Now this is the ...
0
votes
1answer
26 views
Sequences of i.i.d. subgaussian RVs and uniform integrability
Consider a sequence of i.i.d. subgaussian RVs $\{a_{j}\}^{n-1}_{j = 0}$; is $\{a^2_{j}\}^{n-1}_{j = 0}$ uniformly integrable (UI)?
Intuitively it appears to be so; if we take for example $\{a_j\}$ ...
2
votes
1answer
186 views
Period of linear congruential generator
How can you calculate the probability distribution of the period length of a linear congruential generator? That is $X_{n+1} = (aX_n + c) \bmod m$ where $a$ is chosen uniformly at random from ...
0
votes
0answers
29 views
Total set of functions in $L^2(\Omega)$
Are the sets of functions $\{e^{\int_0^T h(s)dB_s -\frac{1}{2}\int_0^T h(s)^2 ds}\}$ and $\{e^{\int_0^T h(s)dB_s}\}$ total in $L^2(\Omega)$? What is the difference? What should one use to prove weak ...
0
votes
1answer
27 views
How to calculate $\Pr[\max(X,Y)<4]$?
Suppose the joint PDF of X,Y is $f(x,y)=1/40$ and $0 < x < 5$ and $0 < y < 8$.
How to calculate $\Pr[\max(X,Y)<4]$?
11
votes
0answers
59 views
Minesweeper - Chance of one-click win
I'd like to know if it's possible to calculate the odds of winning a game of Minesweeper (on easy difficulty) in a single click. This page documents a bug that occurs if you do so, and they calculate ...
14
votes
0answers
103 views
How long does it take a person with this “cheating” data-gathering strategy to achieve a desired result?
I have a perfectly fair coin, and my goal is to prove that it is unfair with a confidence level of 95%. In order to accomplish this, I will cheat. Whenever I fail to have enough evidence, I will ...
1
vote
1answer
34 views
Convergence almost surely
Let $X_n$ and $X$ be random variables. If $X_n \to X$ almost surely, then we have that
$$ \mathbb{P}\left( \lim_{n \to \infty} X_n = X\right) = 1. $$
My question is, can we conclude that
$$ \lim_{n ...
1
vote
2answers
119 views
Question on uniform distribution
Two people agree to meet each other on a particular day, between 5 and 6 PM, They arrive independently on a uniform time between 5 and 6 and wait for 15 mintues. What is the probability that they meet ...
1
vote
1answer
31 views
Counting probability question-what is the sample space in this problem?
Hi folks this is a self learn probability (counting) question from DeGroot. The question is:
Suppose that a box contains r red balls and w white balls. Suppose also that the balls are drawn out ...
1
vote
1answer
47 views
I'm gonna give probability regularization classes
There's this group of high-school level kids that failed probability and they want me to teach them so they can pass that subject, i agreed to be their teacher for this 2 weeks, however i'm not ...
2
votes
1answer
62 views
What is the probability of the number 1 and number 2 employees getting the bonus at a call center?
Two weeks ago, a friend working at a call center told me about their staff bonus policy. Here I paraphrase it.
Suppose employee A answers the maximum number ($N_1$) of calls among the staff, and ...
1
vote
1answer
56 views
$\mathbb E[\frac{\partial}{\partial\theta}\log f(X;\theta)]^2$ and $\mathbb E[\frac{\partial^2}{\partial\theta^2}\log f(X;\theta)]$
$f(x;\theta)=\frac{1}{\pi[1+(x-\theta)^2]}$; $-\infty<x<\infty,\quad-\infty<\theta<\infty$
$\log f(x;\theta)=\log (\frac{1}{\pi[1+(x-\theta)^2]})$
$\Rightarrow \log ...
1
vote
1answer
19 views
Probability element in subset
Let be $A$ a set of naturals numbers from $1$ to $N$. Let be $B\subset A$ with $M=\operatorname{card}(B)$. Is $M/N$ the probability that finds an element belong $M$ choosing randomly any number from ...
0
votes
0answers
47 views
Central Limit Theorem: probability density function of the true mean?
A friend and I are arguing over the meaning of the Central Limit Theorem. I am stating that the normal distribution seen by taking the mean of a large number of samples is a probability density ...
0
votes
1answer
38 views
cauchy schwarz equality: difference in proving style for linear algebra and expectation version
I am interested in proving the following sub version of Cauchy Schawrz equality.
1) LA version :
If $x$ and $y$ are two real vectors and the following holds
$$<x,y> = ||x||.||y||$$ then $x$ ...
0
votes
0answers
34 views
$\operatorname{Prob}\limits_{x,y\in\mathbb{Z}_q^*}[\gcd(xy \bmod q, pq)>1: \gcd(x,pq)=\gcd(y,pq)=1]=?$
For given two distinct odd primes $p$ and $q$, how to count the probability
$$\operatorname{Prob}\limits_{x,y\in\mathbb{Z}_q^*}[\gcd(xy\bmod q, pq)>1: \gcd(x,pq)=\gcd(y,pq)=1]=\, ?$$
2
votes
2answers
29 views
“Expected Probability”: Withdrawing a white ball from a bucket with a random variable X = number of whites
I have a problem with the subject of "Expected Probability" (I don't really know what is the right name for it).
This is an example of a question: (I am not looking for the specific answer, just for ...
0
votes
2answers
42 views
Yet another balls and boxes problem; minimum number of throws so that we have no empty boxes.
I managed to figure out how many empty boxes will be left given n amount of throws, just having a hard time figuring out the minimum number of throws necessary so that we have no empty boxes. Would it ...
1
vote
0answers
34 views
maximize the expected value of the logarithm of the weighted average of random variables
I'm trying to do the following.
$$\max_{m\in\mathbb{R}} \mathbb{E}\left[\log (wA + (1-w)B_m)\right],$$
where $0<w<1$ and $A, B_m > 0$ are correlated random variables. $A$ does not depend ...
0
votes
0answers
10 views
Optimal stopping for random walks
Let $X_0=0, X_1, X_2,\dots, X_N$ be i.i.d. random variables, with Gaussian distribution $\cal N (0,1)$. For $k=0,\dots, N, S_k=\sum_{i=1}^k X_i$ and $Z_k= (N+1-k)S_k^2$. The goal is to get ...
3
votes
1answer
91 views
Balls on Stairs
Recently I realized that lost all computing skill in probability. Please take a look at the following problem.
There are $b$ balls that are thrown one by one and bounce from top to bottom on the $n$ ...
3
votes
2answers
80 views
Speculating on the stock exchange
Imagine you model each stock as a random walk (fractal) and also that you can buy and sell at any price. Suppose also that it 'walks' with the pace of 1.
If you buy, for example, 1000 shares of ...
0
votes
1answer
367 views
Marginal PDF from a joint PDF with an integral that does not converge
I have a joint PDF that has gone through some transformations of
f(x,y) = 12x*(1-y)/y^3 , 0
It definitely is a valid PDF as it has a double integration along its support that equals 1
I am trying ...
1
vote
0answers
29 views
Odds of turning all the columns in solitaire without drawing from the pack
Windows solitaire experiment.
Start a game and see how many cards can be exposed in the columns before a draw from the pack is needed.
(Single or three card game makes no difference here)
As a time ...
2
votes
1answer
60 views
Expected value: Showing $[\Bbb E(X)-\Bbb E(Y)]^2 \geq 2 \cdot \Bbb{Cov}(X,Y)$
The original question is to show that for any Random variables $X,Y$ and $0\leq p \leq 1$
$$p\Bbb V(X)+(1-p)\Bbb V(Y) + p(1-p)[\Bbb E(X)-\Bbb E(Y)]^2 \geq p^2 \Bbb V(X)+(1-p)^2 \Bbb V(Y) +2p(1-p) ...
3
votes
0answers
78 views
A basic doubt on calculating $E[g(X)]$
Assume $X$ is a continuous random variable and $g$ is a measurable function. Then $g(X)$ is a random variable. Now, there are two ways to calculate $E[g(X)]$.
1) Assume $Y=g(X)$ and find the density ...
1
vote
0answers
13 views
Conditions for the ground state of Gibbs ensemble not to be “degenerate”
I am looking at the Wikipedia article on Partition function -- As a measure. Unfortunately the article has no relevant references or reading suggestions.
I am looking for books or other resources ...
1
vote
1answer
196 views
math distribution of averages and distribution of standard deviation for comparison of bytes
Let me change my question:
I have a sequence of 512 numbers (these 512 numbers can have a value between 0 and 255) (in computer mode these numbers can be between 00000000 and 11111111 (bits)). For ...
0
votes
4answers
30 views
Choosing balls out of a bag, replacing them by opposite colour
Can somebody help me out on this question:
A bag contains four white balls and four red balls. A ball is selected at random, removed and replaced by a ball of the opposite colour. A second ball is ...
0
votes
2answers
42 views
Probability that I am not selected in any of 2000 samples?
The population contains 100 million adults, which includes myself. Simple random sampling is used to choose a sample of 1000 adults, 2000 times, independently.
I need to find the probability that I ...
0
votes
1answer
25 views
Neyman-Pearson, solving for $k$
I've got a one major problem using the Neyman-Pearson lemma.
We're testing hypotheses $H_0: \theta \le \theta_0$ vs. $H_1: \theta > \theta_0$. Our $$f(x,\theta) = ...
