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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3
votes
1answer
71 views

Stock market trading / Casino betting / Multi-player fun competition possible with the following input? [on hold]

I would like to program some kind of online betting system for fun. Just for the fun factor, I would like the Twitch chat to be the random input (seed). As can be seen here, you can see one possible ...
1
vote
1answer
28 views

Simple Question about Monotone Convergence Theorem

Suppose we have a sequence of (discrete) random variables $X_0, X_1, \dotsc$ over $E$ and $A \subseteq E$. Let $Y$ be some other random variable. Moreover, let $Z$ be a random variable with values in ...
0
votes
0answers
43 views

Can you verify the combinatoric recurrence?

There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical. ...
0
votes
1answer
19 views

How are Chi Square probabilities calculated?

What steps would one follow to calculate the values in a Chi Square probability table such as https://people.richland.edu/james/lecture/m170/tbl-chi.html? Say you had 15 degrees of freedom and wanted ...
0
votes
0answers
23 views

Is it possible to exchange a sum in a conditional expectation

Let $X_1, X_2, \ldots \geqslant 0$ and $Y$ be RVs over $\mathbb{R}^n$. Then is it true that $\mathbf{E} \left[ \sum_{i = 1}^{\infty} X_i \mid Y \right] = \sum_{i = 1}^{\infty} \mathbf{E} [X_i \mid ...
0
votes
1answer
21 views

Picking a Winner [on hold]

Two friends are playing a game where they try to predict the score when two-sided dice are rolled and the numbers added together. If one of them guesses the score correctly, they get the point. a) ...
0
votes
2answers
49 views

Prove (or disprove) that $\mathbb{E}[X]\geq 0$ for positive random variable.

Let $X$ be a random variable such that $X\in[0,1]$. I was wondering if $\mathbb{E}[X]$ must be $\geq0$. Since $X$ is a positive random variable, we can apply the Markov-inequality: for each positive ...
-1
votes
0answers
27 views

Distribution Problem based on unknown function [on hold]

I got struck at this problems as Function is not given. Any help will be appreciated
1
vote
0answers
19 views

Concentration inequalities for product of gaussians

Are there any concentration inequalities (i.e. probability bounds on how a random variable deviates from its expectation) for the product of $n$ gaussian random variables with zero means and equal ...
-2
votes
1answer
34 views

How to find PDF of ordered random variables? [on hold]

Assumpion: Let $X_1, X_2, \ldots, X_L$ be $L$ independent and identical random variables (RVs). Let $F_{X_i}(x_i)$ and $f_{X_i}(x_i)$ be CDF and PDF of $X_i$. Suppose that $F_{X_i}(x_i) = F_X(x_i)$ ...
1
vote
1answer
29 views

Two contests, an extension of the Coupon Collector's problem

Coupon Collector's Problem Let $X$ be the number of coupons drawn with replacement from an urn containing $N$ distinct coupons until each coupon has been drawn at least once, winning the coupon ...
-1
votes
1answer
27 views

Probability Ques. [on hold]

From previous experience, Bob’s Programming teacher takes down the attendance 40% of the time. Bob’s classmate, Marty, comes late to class (i.e. after the attendance is taken down) 20% of the time. ...
1
vote
2answers
42 views

Uniform distribution and expectation

Let $U \sim \mathrm{Unif}(0,1)$, $X=U^2$ and $Y=e^X$. Compute $E[Y]$ (leave answer as an integral). So essentially we need to compute $E[e^{U^2}]$? I am a little confused how to approach this problem? ...
3
votes
1answer
41 views

Solving a SDE / Finding expectation Value

I am working on a physics problem, and have come across the following stochastic differential equation: $dX(t) = \left( \frac{8}{3} X(t) - 3 X(t)^3\right)dt + dW$. I have tried all the methods to ...
-1
votes
1answer
47 views

Probability and expectation proof

Let X have PDF f(X) and let a,b ∈ R. Show that E[aX + b] = aE[X] + b. I am little confused how do you prove it? Isn't it just the regular proof that the expectation of b is just b?
1
vote
3answers
45 views

Each of two evidences increases prior probability but both decrease it. May this only happen if two evidences are dependent?

I noticed this while working on another problem. My intuition is that the statement is true, but I am not sure. Let A is an event. Evidence 1 and 2 are $E_1$ & $E_2$ correspondingly. $$P(A|E_1) ...
1
vote
1answer
50 views

Using Feynman-Kac, compute the following:

Let $B(t)$ be Brownian Motion and let $\alpha$ be a constant and $T>0$. Compute $\mathbb{E}_{B_{0} = x}\left[\exp\left(-\alpha \int_0^T B(s)^2 ds\right)\right]$. I'm just having a hard time with ...
1
vote
1answer
29 views

Gaussian distribution determined by first two moments

When said that Gaussian distribution is determined by it's mean and variance. How is that different of other distributions? Almost every distribution which I can think of has this property. For ...
1
vote
3answers
54 views

Right answer, wrong explanation, probability of grids?

Two unit squares are selected at random without replacement from an $n\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected squares are ...
2
votes
1answer
43 views

Approximation to a compounded Binomial distribution

I need to find an approximation, from which I can easily sample, to the following compounded Binomial distribution: $X \sim \mathrm{Binomial}(e^{-\epsilon}, \ n)$ where $\epsilon \sim ...
1
vote
1answer
61 views

Can an interval be represented as a set?

In a problem I was asked to to prove the following of a probabilistic model whose sample space is the real line: $$P([0,\infty))=\lim_{n\to \infty}P([0,n])$$ The solution used the previously proved ...
2
votes
2answers
79 views

Find a recursion (combinatorial)

Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. ...
1
vote
1answer
33 views

How does conditional expectation really operate?

Let there be a keyboard with k keys, only 9 of which are numbers, which are 1,...,9. A monkey performs a series of random taps. The series will end as the monkey taps a non-number key. Let $N$ be the ...
1
vote
2answers
34 views

Question about probability distributions

I've recently came across this question: ...
0
votes
1answer
67 views

What is the probability of an event happening in some interval given probability of it in x interval?

Suppose there is an event that happens with a probability of y in x interval of time, what would be the probability of it happening in x/2 interval of time? Would that be y/2 or is there something ...
-2
votes
1answer
41 views

Can someone confirm if the solutions are correct?

For the first one, I did 163 / 1200.. For the second one, I got 24% I think both of mine are correct, but solutions say otherwise.
-1
votes
5answers
58 views

Picking (and replacing) among five balls in an urn [on hold]

An urn contains 5 balls numbered from 1 to 5. A ball is chosen at random and its number is noted the ball is then returned to the urn. this is done a total of 5 times. What is the probability that ...
0
votes
0answers
30 views

How do I calculate conditional PDF?

Obtain $$P(2 < Y < 3 | X = 1)$$ where the joint pdf of X and Y is $$f_{X,Y}(x,y) = (6-x-y)/8$$ where $$0 < x < 2$$ and $$2 < y < 4$$? so first, I did $$f_Y|X=1(y) = ...
1
vote
3answers
68 views

Probability that two numbers differ by one bit

Assuming that t is the bit length of the numbers and that we can pick 2 random numbers (the same number cannot be chosen twice), which is the probability that the two numbers will differ by exactly ...
0
votes
1answer
32 views

overlapping two events in one year with e certain duration. [on hold]

I'm struggling with the following problem: Given that two events are happening in the same year. Event 1 has durantion of two hours, when its happens the duration period is uninterrupted. Event 2 has ...
0
votes
2answers
66 views

Conditional Expectation: Sum inside or outside of expectation?

Let $X,Y$ be some discrete random variables with $Y$ taking values in $\mathbb{N}$ and consider $\mathbb{E}[X]$. Since it is sometimes easier to consider the expectation conditioned on a certain ...
0
votes
0answers
29 views

Interpretation of integral as ratio of joint and conditional densities?

A common exercise in Bayesian statistics is specifying a prior $p(\theta)$ on some parameter $\theta$. We then observe a collection of data $D=(X_1,\dots,X_N)$, the distribution of which is ...
1
vote
2answers
23 views

Find the required Chi-square score for an arbitrarily low p-value (2 degrees of freedom)

I'm trying to use the Chi-Square test to find the significance of data that suffers from the multiple testing problem. Because I have this multiple testing problem, the required p-value to view a test ...
0
votes
0answers
8 views

RBF transformation on a Normally Distributed Random Variable

I have a random vector $\mathbf{X} \sim \mathcal{N}(\mathbf{m,\Sigma})$ which is transformed by a Gaussian Radial Basis Function into the random variable $\mathbf{Y} = K(\mathbf X)$ where $K = ...
-1
votes
1answer
37 views

Basic Probability using combinations [closed]

(a) A committee of 5 people is to be chosen from a group of 10 (6 men and 4 women) (i) How many committees of 5 members can be chosen from 10 people? (ii) How many of the committees from (a) will ...
-3
votes
1answer
21 views

Uniform distribution and real values [closed]

If the random variable $k$ is uniformly distributed in $(0,5)$, What is the probability that the roots of the equation $4x^2+4xk + k + 2 = 0$ are real?
0
votes
2answers
50 views

If a professor has 7 students and they have to at least do 2 assignments each…

The professor has $7$ students. Each student has to do at least $2$ projects. There are $3$ projects: $A, B,$ and $C$. Project $A$ has been assigned $4$ times. $B$ has been assigned $5$ times. $C$ has ...
-1
votes
1answer
34 views

Bayes' theorem with multiple variables

On the page: https://en.wikipedia.org/wiki/Bayesian_inference#Formal_description_of_Bayesian_inference there is the result: $$p(\theta \mid \mathbf{X},\alpha) = \frac{p(\mathbf{X} \mid \theta) ...
0
votes
1answer
27 views

Slow convergence simulating log-normal sample from the normal

I am trying to simulate a log-normal random variable $Y$ with mean $m = \mathbb{E}[Y] = 0.001$ and standard deviation $s = 0.094$ by simulating a normal sample instead, and then exponentiating it. ...
0
votes
1answer
21 views

Finding a moment generating function

I want to find $M(t)$ of $$f(x)= \begin{cases} e^{(-x-1)} & \text{for } x > -1 \\ 0 & \text{otherwise} \end{cases}$$ $e^{(-x-1)}$ I tried to do $$\int_{-1}^{∞ {}} e^{tx} ...
0
votes
2answers
48 views

Chance of failure of a machine in a year - Probability ?(Interview Question)

A machine has 3 components say A,B,C and at any given day chance of failure of any of them is 1%. The machine doesn't work if any of the component fails. So the machine doesn't work if either 1 / 2 / ...
-5
votes
1answer
26 views

Conditional Expectation for IIDs [closed]

What is E[X|X+Y=1] Given X and Y are independent and identically distributed.
-2
votes
1answer
22 views

Probability of a user references in a network [closed]

I am trying to figure out no of possible referrals of a user in a network. Where the size of a network is not fixed but we can set an assumption of 1000 persons. Edit: A user knows few users in a ...
-2
votes
2answers
92 views

What do you call this thing in probability theory? [closed]

I have studied it before but I forgot the name. It is like when the possiblity of something happens is so small, but you created the experience so so many times, then the probability of that thing to ...
3
votes
1answer
18 views

CDF of the difference of two Gaussian mixtures

I have two Gaussian mixtures, $X_D$ and $X_{\overline{D}}$: $$ f(X_D) = \sum_{c=1}^m f(X_D\mid C=c)P(C=c) = \sum_{c=1}^m \phi(x-\mu-g(c))P(C=c), $$ $$ f(X_\overline{D}) = \sum_{c=1}^m ...
2
votes
1answer
80 views

Weird Induction…?

I was watching this video earlier and I couldnt figure out why the following step was possible. This is the original problem: $\sum_{i = 0}^{n} \binom{n + i}{i} = \binom{2n + 1}{n + 1}$ At one ...
0
votes
0answers
46 views

probability problem in a die game

I'm stuck with the following question: A, B and C are playing a game. At each turn, everyone tosses a fair die and the one with the largest number takes one dollar from the one with the least ...
0
votes
1answer
25 views

Probability the range is disjoint

Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is ...
2
votes
1answer
49 views

Sum of squares of terms of a binomial expansion

I have a coin that show heads with a probability $p$. I toss it $N$ times and count the number of heads. I repeat the experiment once more. What's the probability that I get the same number of heads ...
2
votes
0answers
20 views

Tail field versus germ field of Brownian motion

Continuing my foray into Brownian motion (apologies for the bombardment...), I'm trying to verify the details of a proof of Durrett of the following 0-1 property of the tail $\sigma$-algebra of ...