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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2
votes
1answer
63 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. ...
0
votes
2answers
43 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 ...
5
votes
1answer
360 views

Probability that a given Poisson variable samples greater than its mean $\lambda$, provided $\lambda > D$

Given a random variable $X \sim \text{Poisson}(\lambda)$ such that $\lambda > D$, with $\lambda, D \in \mathbb{N}$, what is the probability that a sample obtained from $X$ is greater than ...
-2
votes
1answer
14 views

How to find PDF of ordered random variables?

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 $F_{X_i}(x_i) = F_X(x_i)$ ...
2
votes
0answers
140 views

Building Bayesian Networks, Causality and Cyclic Reasoning

I am studying Bayesian Statistics and I am trying to get a good understanding on Bayesian Networks, which seems to be vital in order to make something useful in Machine Learning. Most of the texts I ...
2
votes
4answers
95 views

When will Andrea arrive before Bert?

The question was as follows- on any given day, Andrea is equally likely to clock in at work any time from 8:50am to 9:06am. Similarly, Bert is equally likely to to clock in at work at any time ...
-1
votes
1answer
43 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
4answers
159 views

How to understand $E(XY)$ intuitively

I have no trouble understanding $\displaystyle E(X)=\int xf(x)\,dx $ and $\displaystyle E(Y)=\int y f(y)\,dy$ As each $x$ multiplies the corresponding $f(x)$ and we take the integral of it to ...
-1
votes
0answers
20 views

Distribution Problem based on unknown function

I got struck at this problems as Function is not given. Any help will be appreciated
0
votes
1answer
597 views

Expected state of a Markov chain

Let's start with a slightly trivial Markov chain defined as follows: the beginning state is called $1$ and the set of states is $\mathbb{N}$. At each step, when the current state is $n$, the ...
-1
votes
1answer
26 views

Probability Ques.

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
0answers
41 views

Find formula that produces desired graph

Let's say we have an urn of balls of size $n$, Each ball has probability $p$ of being red. I take a sample from this urn without replacement and calculate the probability of having at least a ...
-2
votes
1answer
39 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
vote
1answer
33 views

Using Feynman-Kac, compute the following:

Let $B(t)$ be Brownian Motion and let $\alpha$ be a constant and $T>0$. Compute $\mathbb{E}_{B_{0} = x}[ exp(-\alpha \int_0^T B(s)^2 ds)]$ I'm just having a hard time with this one, any help?
-1
votes
5answers
48 views

Picking (and replacing) among five balls in an urn

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 ...
1
vote
3answers
50 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 ...
0
votes
0answers
16 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
vote
3answers
41 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
2answers
31 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? ...
2
votes
0answers
21 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 ...
0
votes
1answer
23 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 ...
3
votes
2answers
1k views

Flipping Cards Probability

You have a deck of cards, 26 red, 26 black. These are turned over, and at any point you may stop and exclaim "The next card is red.". If the next card is red you win £10. What's the ...
2
votes
1answer
27 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
559 views

Solving Probability Density Function for continuous random variable

The probability density of a random variable $x$ is $$f(x)=a\ \cdotp x^2\ \cdotp \mathrm{e}^{−kx}\ (k>0,\ 0\leq x\leq \infty)$$ Then, the coefficient $a$ equals $$(i)\frac{k^3}{2}\ \ \ \ (ii)\ k^3 ...
1
vote
1answer
19 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 ...
15
votes
3answers
2k views

Choose a random number between 0 and 1 and record its value. Keep doing it until the sum of the numbers exceeds 1. How many tries do we have to do?

Choose a random number between 0 and 1 and record its value. Do this again and add the second number to the first number. Keep doing this until the sum of the numbers exceeds 1. What's the expected ...
1
vote
1answer
56 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 ...
1
vote
1answer
32 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 ...
0
votes
1answer
36 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 ...
1
vote
2answers
32 views
3
votes
1answer
197 views

Relative entropy between singular measures

Usually, to define relative entropy between two probability measures, one assumes absolute continuity. Is it possible to extend the usual definition in the non absolutely continuous case?
1
vote
3answers
65 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 ...
-1
votes
1answer
31 views

Does these inequalities hold in General for probability distribution? [on hold]

Let $Q(y)$ be a probability density of $y \in [-1,1]$. Then for $t> 0$, the inequalities are $\displaystyle \int_{0 \leq y <t} y^2 Q(y) \, dy \leq t^2 \int_{0 \leq y <t} Q(y) \, dy $. ...
0
votes
1answer
31 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
0answers
27 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) = ...
0
votes
2answers
26 views

Conditional Expectation: Sum inside or outside?

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
1answer
22 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. ...
3
votes
1answer
467 views

Recurrence Relation, Discrete Math problem(Homework)

There is a disk, separated into n sections, as indicated in the graph. For each section, you can paint it with one color out of four: Red, Yellow, Blue, Green. The rule is adjacent sections can't have ...
2
votes
3answers
89 views

High computation in probability

Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at ...
0
votes
0answers
23 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 ...
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
37 views

Basic Probability using combinations [on hold]

(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 ...
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 = ...
0
votes
1answer
408 views

Combination Problem Confusion

I'm reading my Probability + Stats book for engineers and came across this example in the actual reading. The solution is given in the reading...but I don't understand why we have to divide in the ...
-1
votes
1answer
33 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) ...
-3
votes
1answer
20 views

Uniform distribution and real values [on hold]

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
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
44 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 / ...
1
vote
1answer
36 views

How is Poisson Distribution simply discerned? How is it related to the Binomial distribution?

There is this question which I thought I had understood, until taking a look at the answers: Let a floor tile be composed of different four tiles: a black one of size $1\times1$, a red $3\times 3$ ...
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 ...