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
41 views

Show martingale space is a Banach space

Let $\mathcal{H}^1 = \{M \in \mathcal{M}, E[sup_{t\geq 0} |M_t|] < \infty\}$, where $\mathcal{M}$ is the space of right continuous with left limits martingales. Show that $\mathcal{H}^1$ is ...
0
votes
1answer
21 views

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$ i) Find the regions where the joint pdf of $(Z,W)$ is positive. ii) Find the ...
0
votes
0answers
13 views

The bound of Rayleigh quotient

Suppose $A$ and $B$ are some random positive definite matrix (e.g., covariance matrix) draw from some distribution. And I want to compute ...
1
vote
0answers
56 views

Probability - Hausdorff distance

On a 4x4 table 2 coins with radius 1 are thrown.They may overlap partially or completely but remain completely on the table.What is the probability that Hausdorff distance between the two nickels to ...
0
votes
1answer
21 views

An example of covergence to an exponential distribution, the role of continuity

I got a probability problem I can solve, but my solution does not use an assumption which is given in the formulation of the problem. I am afraid that this is might be a sign that my solution is ...
0
votes
2answers
32 views

Why would a uniform prior distribution give a different result than a purely frequentist approach?

I would expect a uniform prior to be a good example of an uninformed prior and get the same result as the frequentist approach. However, this is not the case. As an example, let's look the classical ...
1
vote
1answer
20 views

Showing independence of two random variables

The problem is here The trouble im having is showing how $\bar{x}-\bar{y}$ is independent of $S_{pool}$. I know the covariance of ( $\bar{x}-\bar{y}$,$X_i-\bar{x}$)=0 and similarly for the other ...
0
votes
1answer
23 views

Exponential Random Variable question regarding property

I am fairly new to this site. Below I show my question. Consider $T$ an exponential random variable. Show $$\mathbb{P}[T>s+t|T>s] = P[T>t]$$ I use Bayes rule ...
1
vote
0answers
16 views

distribution of the maximum of independent poisson random variables.

Let $X_i$ $i=1,\dots,n$ be independent poisson random variables with $X_i \sim \text{Poisson}(\lambda_i)$ then we define $X = \max_i X_i$ how does $X$ distribute? Is easy to see that ...
0
votes
0answers
14 views

Variance of event counting

I have this question (not homework, review problem for qualifying exam), tried approaching it a couple of ways (unsuccessfully). Any recommendations? Let $X_1,..,X_n$ be i.i.d continuous rvs. A ...
1
vote
1answer
12 views

Simple multivariate probability

$$F(y_1,y_2)=1, 0 \le y_1 \le 1 , 0 \le y_2 \le 1$$ $$F(y_1,y_2)=0, elsewhere$$ Find $$P(y_1+y_2) \le \frac{5}{4}$$ I can solve this geometrically and I know the answer is $\frac{23}{32}$ I ...
-2
votes
1answer
10 views

If a dice is roll three times and each number noted (i.e. 543) what is the probability for the combination to occur [closed]

If a die is rolled 3 times and each number noted ( for example 5 4 3) what is the probability for that combination to occur?
0
votes
1answer
13 views

Distribution of linear combination of iid exponential rv

If $X_1,X_2,\dots,X_n$ are i.i.d exp$(\lambda)$. How can I find the distribution of $U_n = \sum^n_{i=1} X_i/i$? Is this CF, MGF, PGF related? Thanks.
0
votes
0answers
15 views

The distribution of the sum of a uniform random variable and a binomial random variable

I'm asked to find the distribution of $U=X+Z$, where $X\widetilde~R(0,1)$ - That is, $X$ has a uniform distribution for $x\in]0;1[$ $Z\widetilde~bin(1,1/2)$ - That is, $Z$ has a binomial ...
0
votes
1answer
29 views

How much more likely is X% than Y%? [closed]

77% of group A like singing 52% of group B like singing How much more likely is group A to like singing than group B? And can I say it like 'twice as likely'?
-3
votes
0answers
24 views

Problem with Poisson distribution [closed]

Let $x$ be the numbers of cars passing per minute through a point between 8 AM and 10 AM on Sunday with a mean of $5$. Find the probability of observing $3$ or fewer cars during any given minute?
1
vote
1answer
16 views

Do the expressions for conditional probability with unions expand like this

Is it correct that: $$\Pr[A \mid B \cap C \cap D] = \frac{\Pr[A \cap B \mid C \cap D]}{\Pr[ B \mid C \cap D] }$$
0
votes
0answers
10 views

Wasserstein distance and maximization covariance

My question deals with the second order wasserstein distance $W_2$ on the set of measures, which is defined by: $W_2(\nu_1,\nu_2)^2= inf_{\Pi(X,Y)} E_{\Pi} (X-Y)^2$ where $\Pi$ is chosen such that ...
0
votes
1answer
34 views

Setting bound for an infinite expected value

Say $X=2^Z$ and $Z$ is a geometric random variable with $p=1/2$. It follows that, $E[X] = \infty$ So setting the upper bound by the markov inequality, $$P(X \geq t) \leq \frac{E[X]}{t} = ...
0
votes
1answer
43 views

The probability of exactly $r$ guests leaving with their own hats after a random permutation

Problem Arriving at party $n$ guests throw their hats into pile. When they leave they each take a hat that is chosen randomly from the pile. We want to compute the probability of the event that ...
1
vote
1answer
29 views

Combinations with Repetition

I am looking the basics of combinations with repetition. The other name is Stars and Bars problem. On MIT OCW I found this: An ice-cream store specializes in super-sized deserts. They offer a ...
0
votes
2answers
25 views

What's an intuitive description of the meaning of standard deviation in a discrete uniform distribution?

Just starting out with distributions, so I'm looking for an every day explanation to help me understand. I've read that for a discrete uniform distribution, the standard deviation is a measure of the ...
0
votes
0answers
12 views

Probability - Wind speed calculation

I've been asked to find the probability of wind speed exceeding 15 m/s in a certain area. The question states that a bridge must be shut down if the wind exceeds that value and so my challenge is to ...
0
votes
0answers
17 views

A two-stage experiment where the first stage has two independent outcomes

If $P(Y_1\in \cdot|X_1, X_2) = P(Y_1\in \cdot|X_1)$ and if $P(Y_2\in \cdot|X_1, X_2) = P(Y_2\in \cdot|X_2)$ and if $X_1$ and $X_2$ are independent, are $Y_1$ and $Y_2$ independent given $X_1, X_2$, ...
1
vote
2answers
34 views

Simple convolution problem

Let $X$ be continuous uniform over $[0,2]$ and $Y$ be continuous uniform over $[3,4]$. Find and sketch the PDF of $Z = X + Y$, using convolutions. So I have: $$f(x) = 1/2, 0 \leq x \leq 2 $$ $$f(y) ...
0
votes
0answers
30 views

What is the probability of shooting a puck overlapping the boundaries to get a prize?

Hello, I am new to the forum, and the maths teacher just asked the whole class this question about probability and all of us can't answer it. The question is: there are 9 grid squares on the table, ...
0
votes
1answer
17 views

Covering deficits with values with different weights

SO I have a couple of assessments with specific weights as follows: Assignment 1: 5% => Mark 60% Assignment 2: 5% => Mark 53% Assignment 3: 5% Assignment 4: 5% Test 1: 30% => 47% Test 2: 30% ...
1
vote
3answers
44 views

Probablity in dice game [closed]

In a game of dice 5 dice are rolled simultaneously find the probability that the roll produced least four of a kind. please give me the solution.
0
votes
0answers
14 views

Characterstic Functions and Recovery

Assume that I have a pdf, call it $f$, that is supported on $[0,2]$. Let $\varphi(t)$ be the corresponding characteristic function, which is known to me. Is there some common method to recover the pdf ...
-2
votes
1answer
63 views

Given $B \cup A = B$ and probability and set theory axioms, prove $\mathbb{P}(A) \leq \mathbb{P}(B)$.

I need to prove that $\mathbb{P}(A)$ is less than or equal to $\mathbb{P}(B)$ using only this three things: $B \cup A = B$ The three axioms of probability: a) $\mathbb{P}(A)$ is greater or equal to ...
0
votes
1answer
33 views

Conditional probability of an inspector having prior training, given the failure to detect a weapon

Ninety percent of new airport-security personnel have had prior training in weapon detection. During their first month on the job, personnel without prior training fail to detect a weapon 3% of ...
0
votes
1answer
39 views

Chance of an accident (conditional probability)

An insurance company has 50% urban and 50% rural customers. If every year each urban customer has an accident with probability μ and each rural customer has an accident with probability λ. Assume that ...
2
votes
1answer
50 views

can probability be negative

I am solving a question that says: Given that $$P(B)=P(A\cap B)=\frac{1}{2}$$ and $$P(A)=\frac{3}{8}$$ Find $$P(A\cap {B}^{c})$$ My answer is $$P(A\cap {B}^{c})=P(A)-P(A\cap B)=(-)\frac{1}{8}$$
0
votes
1answer
11 views

What are the odds of two values occurring in a specific order.

So, I have a playlist including 33 songs by one of my favorite artists. Now I know that the "shuffle" feature isn't truly random. But, upon listening to the playlist, two songs "Here We Go" and "Here ...
0
votes
0answers
18 views

Proper definition of Kullback-Leibler divergence for densities.

Let $f$ and $g$ be densities on the same set $X$. The Kullback-Leibler divergence is expressed by this famous formula: $$ D(f||g) = \int_{X^{+}} f(x)\log\left(\frac{f(x)}{g(x)}\right)dx \textrm{,} $$ ...
0
votes
0answers
17 views

The random variable $Y$ is distributed with a mean of $15.5$ and variance of $3.62$. How to find the probability $\mathbb P(15.2 < Y < 16.8)$? [closed]

The random variable $Y$ is distributed with a mean of $15.5$ and variance of $3.62$. The probability $\mathbb P(15.2 < Y < 16.8)$ is...? How do you do this if you don't know that the ...
1
vote
1answer
36 views

mutually exclusive and independent for two dice problem

i'm working on this problem and I'm not sure if I did it correct The question is, a random man rolls 2 dice. (a)Sum = 5 (b)first die is 4 (c)sum = 7 (d)two dice have same # So I drew a 6x6 graph ...
1
vote
3answers
26 views

Unconditional probability [closed]

I understand the tree diagram but my answer is wrong. Urn I contains three red chips and one white chip. Urn II contains two red chips and two white chips. One chip is drawn from each urn and ...
0
votes
1answer
22 views

Conditional Distributions

Choose a random integer $X$ from the interval $[0, 4]$. Then choose a random integer $Y$ from the interval $[0, x]$, where $x$ is the observed value of $X$. Make assumptions about the marginal pmf ...
2
votes
1answer
25 views

Product of two Gaussian PDFs is a Gaussain PDF, but Produt of two Gaussan Variables is not Gaussian

The Product of Two Gaussain Random Variables is not Gaussian distributed: Is the product of two Gaussian random variables also a Gaussian? Also Wolfram Mathworld So this is saying $X \sim N(\mu_1, ...
0
votes
1answer
23 views

show $P(B^c\mid A) < P(B^c)$

If $P(B\mid A) > P(B)$, show that $P(B^c\mid A) < P(B^c)$. I have started this problem by adding $P(B^c\mid A)$ to both sides of the given inequality. $$P(B^c\mid A) + P(B\mid A) > ...
1
vote
1answer
26 views

What is the probability of rain over a number of days, given the probability of rain for each day?

Assume that meteoroligists can predict the probability of rain accurately. If they predict the next seven days all have a 20% chance of rain, what is the probability of it raining at least one of ...
0
votes
0answers
35 views

How can I find the PDF for this random variable? [closed]

Could you please give me a hint to solve this problems? Assuming $X_i$ and $Y_i$ , $i=1,\ldots,N$ are exponential random variable with mean value $\Omega_X$ and $\Omega_Y$. We define a new random ...
-1
votes
0answers
15 views

Is summing posterior probabilities to obtain total counts valid?

I'm using a Bayesian approach to classify my data set into two mutually exclusive groups. From what I read typically one would apply a decision rule to add a count into group 1 if $P_{g1}>P_{g2}$ ...
0
votes
1answer
25 views

Marginal Density Function from Variance

I have an assignment for a lesson: In general, if E(X2)=[E(X)]2=µ2, determine fX(x) I think he made a mistake. E(X2)=[E(X)]2 should be E(X2)-[E(X)]2. Am I right and how can I determine fX(x)
1
vote
0answers
58 views

Expected number of sides of a die which do not show up by 6 throws

Let $X$ be the number of sides of a die that do not show up by 6 throws. I want to calculate $E(X)$. I think the calculation has to look something like this: $ E(X) = 0 \cdot P(X=0) + 1 \cdot ...
0
votes
0answers
22 views

Inequality with cumulative probability function of binomial distribution

Prove that $F(k; 2k+1, p) > F(k-1; 2k-1, p)$ where $p < 1/2$. Here $F$ is the cumulative probability function of binomial distribution Intuitively the inequality is obvious as expected ...
1
vote
2answers
33 views

expected number of defective robots

Factory produced $n$ robots. Every of robot is defective with probablity $q$. If robot is defective then tester detect it with probablity $p$. We assume that tester detected $Z$ defective robots. Let ...
1
vote
1answer
11 views

Sequence of Gamma r.v.s converges in probability to 1

Let $\{X_n\}$ be a squence of Gamma distributed random variables with pdf $$ f(x;\alpha,\beta) = \begin{cases} \hfill \dfrac{x^{\alpha - 1}e^{-x/\beta}}{\beta^{\alpha}\Gamma(\alpha)} ...
1
vote
0answers
22 views

Probability of Topological space [closed]

Let $|S|=n$. What is the probability of a randomly chosen collection of subsets of $S$ forms a topological space?