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 ...
0
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2answers
21 views
Conditional Probability Problem
An insurance company examines its pool of auto insurance customers and gathers the following information:
(i) All customers insure at least 1 car
(ii) 64% of all customers insure more ...
0
votes
0answers
12 views
B.F.'s generated by disjoint subfamilies are independent
Problem 5 of Section 3.3 from "A Course in Probability Theory" by Kai Lai Chung
If $\{X_\alpha\}$ is a family of independent r.v.'s, then the B.F.'s generated by disjoint subfamilies are independent.
...
0
votes
0answers
31 views
Product of Uniform and Gamma Random Variables
Let $X\sim\operatorname{Gamma}(1+\alpha,1)$ and $U\sim \operatorname{U}[0,1]$ be independent, $\alpha < 1$
How do you go about proving that $XU^\frac{1}{\alpha}\sim\operatorname{Gamma}(\alpha,1)$?
...
0
votes
1answer
20 views
Examples of convergence of random variables
First, let's recall the definitions of 4 different types of convergence:almost surely, in $r$th mean, in probability and in distribution:
$X_n\xrightarrow{a.s.}X$ if $\{\omega \in ...
0
votes
1answer
20 views
Find distribution with probability generating function
The probability generating function of $X$ is $G_x(s)=\frac{1}{2}(s^9(1+s^2))$. Find $EX$ and probability distribution function.
$EX=G_x^{'}(s)=\frac{1}{2}(9s^8+11s^{10})$
How about pdf? Do I need ...
2
votes
1answer
28 views
This special random subset of uniformly distributed numbers is still uniformly distributed?
I asked similar question in A special random subset of uniformly distributed numbers is still uniformly distributed?
Here, I slightly change my random number generation method, and want to see ...
0
votes
1answer
33 views
Probability exponential distribution.
May I please borrow your expertise or could anyone check if I'm on the right track please?
Consider customers arriving at a bank. The bank has $2$ types of customers - business and personal. On ...
1
vote
0answers
58 views
Probability drawing two cards
I am trying to set up a probability table for the events of drawing two cards from a $52$ card deck. What counts is either an exact match or a match in flush with two already drawn cards from another ...
1
vote
2answers
19 views
Closed form for Exponential Conditional Expected Value & Variance
I am wondering if there is a closed form for finding the expected value or variance for a conditional exponential distribution.
For example:
$$ E(X|x > a) $$ where X is exponential with mean ...
0
votes
1answer
18 views
Showing uniform convergence in probability
Suppose you want to show $sup_{x\in D}|f_n(x)|\to_p 0$, for $n\to \infty$, where $D\subset \mathbb R$ is a compact interval, $f$ is continuous depending on one or more random variables, and $\to_p$ ...
0
votes
1answer
18 views
Convergence in $L^p$ and $L^q$ - multiplication
We have: $X_n \rightarrow X$ in $L^p$ and $Y_n \rightarrow Y$ in $L^q$. Moreover $p,q>1$ are such that $\frac{1}{p} + \frac{1}{q} =1$. Prove that $X_nY_n \rightarrow XY$ in $L^1$. Please, can you ...
-2
votes
0answers
38 views
Uniform probability question
Anyone here that can solve this challenging question that I have?
Let $U \sim U[a,b]$. Suppose $X = U$ and $Y = \frac{1}{2} U$.
Find $P(X \le x, Y \le y)$ for $-\infty \le x, y \le + \infty$.
0
votes
0answers
46 views
Probability exponential distribution question
Could anyone please help me answer these questions? Or a little hint as to how I can answer them? It's for my assignment that's due tomorrow.. Really appreciate if anyone could help!
Consider ...
0
votes
1answer
35 views
Rolling dices and simple problem
I'm facing the following problem.
Let's say I have N dices in a hand. I need to calculate how much time I should roll my dices to make all of them equal selected (pre-defined) number. Each time when ...
2
votes
1answer
29 views
probability of getting a double six (2 dice) rolling them 24 times
This is what I got. 1/6 * 1/6 = 2.78% * 24 = 66.72%
I believe that since it is a six sided dice, since you roll both of them simultaneously it would be (1/6)*(1/6).
So since they are rolling them ...
2
votes
1answer
46 views
Hard Probability Inequality
I am new in this forum and I am happy to find it, because it seems a very precious place for asking questions.
My question is about some probability inequality. I formulate this as following.
Let ...
1
vote
1answer
55 views
The man with two boys [duplicate]
I have recently seen a probability question which says
"i am asking randomly the persons I met if they are having two chidren and one of them is a boy who was born on tuesday. At last I met one whose ...
1
vote
1answer
16 views
Calculating the probabilities of different lengths of repetitions of X length numbers
I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't ...
0
votes
1answer
18 views
Independence of transformed variables
There are two independent variables X and Y. Y is an input for non deterministic algorithm f, and the output of f(Y) is Z. How to prove that X and Z are independent?
0
votes
0answers
28 views
Convergence and properties of a random permutation.
I'd be interested in the answer, but I think this is more of a challenge question than anything else. I haven't really decided whether to have a serious attempt at solving it myself.
Suppose I have ...
0
votes
1answer
16 views
Expectation of function of stochast
I've got a general question regarding a certain sticking point I often encounter. When tackling questions where for example an UMVUE (uniformly minimum-variance unbiased estimator) has to found I get ...
0
votes
1answer
31 views
Conditional Probability: Bayesian Cause/Effect Question
The probability that a randomly chosen male has a circulation problem is 0.25. Males who have a circulation problem are twice as likely to be smokers as those who do not have a circulation ...
1
vote
0answers
33 views
Urns version of Laplace's law of succession
I'm trying to prove urns version of Laplace's law of succession my professor suggested. Laplace's law states that the chance that the next trial is a success given $j$ successes out of the first $n$ ...
13
votes
1answer
122 views
Generalized nontransitive dice
Let $X_1, \ldots, X_n$ be a collection of random variables. Consider the directed graph with vertex set $\{ 1, 2, \ldots, n \}$ where there is a directed edge $i \to j$ if $\mathbb{P}(X_i > X_j) ...
0
votes
1answer
52 views
Probability of two random n-digit numbers dividing each other
Let $n$ be a positive integer. Suppose $a$ and $b$ are randomly (and independently) chosen two $n$-digit positive integers which consist of digits 1, 2, 3, ..., 9. (So in particular neither $a$ nor ...
1
vote
0answers
34 views
Fast way to estimate cardinal number of subset
I have a large set $S$ of items, but the set is not exactly known. All I know are the cardinal numbers of categories i.e. a number of disjoint subsets,
$ \vert{S_1}\vert \dots \vert S_n\vert$ with ...
0
votes
0answers
35 views
Estimating the radius of a circle
I have a circle iwth radius $r$. I want to test the hypothesis that $r \leq 2$ vs. $r >2$ based on the posterior of $r$. $r$ follows the prior distribution: $f(r) = \frac{2}{r^{2}}$, $ r >0.5$. ...
1
vote
1answer
33 views
What kind of functions can be moment-generating functions for a random variable?
Given an infinitely differentiable function $ g: \mathbb{R} \rightarrow \mathbb{R}$, can we always find a distribution function $f_X$ of some random variable $X$ so that
$g(t) = \int_{-\infty}^\infty ...
2
votes
0answers
21 views
Neyman-Pearson lemma on Normal distribution
We've got a random sample of iid $X_1,\dots,X_n$. We're testing the mean of $X \sim \mathcal{N}(\mu,\sigma^2)$, where $\sigma^2$ is known. The size of the test $\alpha=0.05$.
$H_0: \mu=0$
$H_1: ...
1
vote
0answers
55 views
How does this violate probability theory?
Given: $X = Y^2 + Z^2$ (hence $E[X] = E[Y^2] + E[Z^2]$)
$p(X = 1) = .52$, $p(X = 4) = .24$, $p(X = 16) = .24$
$p(Y = -1) = .5$, $p(Y = 3) = .5$
Question: Despite not being handed any information ...
1
vote
1answer
27 views
Convolution of r.v.'s
Suppose a sequence $\{X_{n} \}$ of pairwise independent r.v.'s. If $F_{X_n}$ is the distribution function of $X_n$ then $ F_{X_1 + \ldots + X_{N}} = F_{X_1} \ast F_{X_2 + \ldots + X_{N}}$ ...
1
vote
1answer
17 views
Conditional joint probability and independence
Let's have a joint probability of three events, $\mathbf{P}(X,A,B)$. If $\mathbf{P}(X|A) = \mathbf{P}(X)$, can we show that $\mathbf{P}(X|A,B) = \mathbf{P}(X|B)$? If so, how?
-1
votes
0answers
45 views
$P\{X_t=-X_t \}=1$
If we define that $X_t$ is Brownian motion over space $(\Omega,\mathcal F ,\mathcal F_t;P) $,
then why is it true that the fact that $X_t$ is Brownian motion implies that $P\{X_t=-X_t \}=1$ is ...
1
vote
2answers
27 views
Uniform distribution on the n-sphere.
I have the next RV:
$$\underline{W}=\frac{\underline{X}}{\frac{||\underline{X}||}{\sqrt{n}}}$$
where $$X_i \tilde \ N(0,1)$$
It's a random vector, and I want to show that it has a uniform ...
1
vote
1answer
17 views
Joint distribution of multiple binomial distributions
In the picture below, how do they arrive at the joint density function? I understand how Binomial distributions work, but have never seen the joint distribution of them.
The original file can be ...
0
votes
1answer
20 views
Probability of all elements of a subset being coprime
Let $S=\{1,..,n\}$ and $R \subset S$ ($|R|=k$, $k<n$) -- $R$ is a random subset of $S$. Let $m=min(R)$, and $R'=\{x-m: x \in R, x \neq m\}$, so $|R'|=k-1$.
What's the probability that ...
2
votes
3answers
89 views
Compute value of $\pi$ up to 8 digits
I am quite lost on how approximate the value of $\pi$ up to 8 digits with a confidence of 99% using Monte Carlo. I think this requires a large number of trials but how can I know how many trials?
I ...
0
votes
0answers
11 views
expectation of logarithm under generalised inverse gaussian
I want to follow the following integral:
$$\frac{1}{C}\int_0^\infty \log(z)\,z^{p-1}\exp\left(-\frac{az+b/z}{2}\right)\,dz$$
where C is the normalising constant.
The following might be useful ...
0
votes
1answer
25 views
Proof of Bienayme Inequality
I have a bit of trouble about the proof of Bienayme Inequality.
Bienayme Inequality is as follows:
If X has mean $\mu$ and variance $\sigma^2$, then
$$\mathbb{P}\left(\frac{|X-\mu|}{\sigma}\ge ...
1
vote
0answers
13 views
Dimension free Concentration bounds for Martingales
Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$:
At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...
2
votes
0answers
41 views
Probability distribution for a digit of a number
If someone choose a digit $\alpha$ and a digit $\beta$ independently. Each one can be in $0,1, ...,9$. So $\mu = \alpha \beta$ (e.g. if $\alpha = 5$ and $\beta = 3$ then $\mu =53$). And I observe a ...
0
votes
1answer
31 views
Continuous Non negative martingale converging to 0
Is there any (non trivial) continuous non negative martingale which converges to 0?
1
vote
1answer
25 views
Probability of catching subway.
A blue tram shows up randomly in a uniform distribution given any hour of the day at a certain stop. A person shows up independently within this same hour. If they are only willing to wait 10 minutes ...
2
votes
1answer
34 views
Finding a PDF from a function
I have a function $y = f(x),\ x\in\mathbb{R}$ (assume $f(x)= \sin(x)/x$ if you need an example). How can I find the probability distribution function (PDF) of $y$, assuming $x\sim U(\mathbb{R})$ ...
7
votes
2answers
64 views
Find: The expected number of urns that are empty
A total of $n$ balls, numbered $1$ through $n$, are put into $n$ urns, also numbered $1$ through $n$ in such a way that ball $i$ is equally likely to go into any of the
urns $1, 2, . . . , i$. Find ...
0
votes
0answers
25 views
Probability that a sub-sequence of i.i.d. zero-mean Gaussians is closer to a given point than the origin
I am given a sequence $X=\{X_1,X_2,\ldots,X_n\}$ of $n$ i.i.d. zero-mean Gaussian random variables $X_i\sim\mathcal{N}(0,\sigma^2)$, and a vector $\mathbf{y}=\{y_1, y_2, \ldots, y_m\}$ of $m$ real ...
0
votes
1answer
31 views
Approximation of a random variable by a sequence of simple random variables
It said in a probability book that any non-negative random variable $X$ can be approximated by a sequence of simple random variables (finite range) $X_1,X_2,\dots,X_n$ such that ...
-1
votes
0answers
29 views
markov question
i have this question pleas consider it
is it possible for me a bit more clear in solution or link to any note from this section of markov chain
The markov chain in continuous time has state space ...
1
vote
3answers
40 views
A simple probability reasing to predict rain fall
A friend told me the following about whether it will rain tomorrow (or not):
The probability that it will rain tomorrow is $1/2$ since it will either happen or not. But -even as a non mathematician- ...
2
votes
1answer
39 views
Conditional Probability, Lack of Dependence on a Parameter
I am trying to understand why the following is true:
$$
p(f(Y) = f(y) \mid Y = y) = p(f(Y) = f(y) \mid X = x, Y = y) \qquad \ldots \text{(Eq. 1)}
$$
where $Y$ and $X$ are random variables, and $f(Y)$ ...
