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Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

5
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1answer
602 views

Probability that, given a set of uniform random variables, the difference between the two smallest values is greater than a certain value

Let $\{X_i\}$ be $n$ iid uniform(0, 1) random variables. How do I compute the probability that the difference between the second smallest value and the smallest value is at least $c$? I've messed ...
0
votes
1answer
67 views

Probabilty Combinatorial related; The Messy Mail Man

Given the messy mail man problem of n pieces of mail, and X RV which value is "how many pieces of mail arrived to the right mail box," how do I compute E(X)? I saw a solution using indicators, but I ...
1
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1answer
350 views

The Nature of Probability Mass/Density Functions

Consider a certain random variable and all its possible probability mass functions (or probability density functions). What structure does this space have? For example, it can be endowed with a ...
3
votes
2answers
1k views

Interpretation of a probability problem: expected value.

I am having a few doubt on the interpretation of this problem that I have read on book about interviews questions. Here the text: A mythical city contains N=100,000 married couples but no children. ...
0
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2answers
201 views

Distribution of a continuous random variable

I was reading through my"Random Variability in Business Situations" notes and wanted to enquire about some difficulty I've encountered. On the fourth and final column "probability calculated from ...
2
votes
1answer
747 views

Mean excess life conditionally on current life in a renewal process

I am self-studying renewal processes and I came across an interesting problem. How would one go about finding the expectation of excess life given that current life is equal to x? I am assuming in ...
-1
votes
2answers
149 views

MLE of parameter in distribution of gaussian time series [closed]

suppose $X_1,X_2,\ldots,X_n$ have joint distribution. if $X_1\sim\mathcal{N}(0,1)$ and for $j=0,1,\ldots,n-1$ conditional distribution $X_{j+1}|X_1=x_1,\ldots,X_j=x_j\sim\mathcal{N}(\rho x_j,1)$, then ...
0
votes
2answers
656 views

Probability Combinatorial related; rolling dice

when 3 fair dices roll, let Y be a Random Variable of which value is the minimum result of all 3 dices results. for example if the results are (2,3,2) then Y=2. how do i find, in the most elegant ...
4
votes
1answer
4k views

coin flips and markov chain

Consider the case of an infinite (or finite $n$) string of coin tosses, and let $q$ and $1-q$ be the probabilities that the coin comes up tails and heads, respectively. (For simplicity, we can take $...
1
vote
1answer
622 views

Multinomial distribution: probability that at least one variable takes a certain value

Let $(X_1,\ldots,X_M)\sim \operatorname{Mult}(N;p_1,\ldots,p_M)$ follow a multinomial distribution. What is the probability that at least one of the variables takes a certain value, i.e. $\mathbb{P}(\...
2
votes
1answer
213 views

Rolling dice, fixed number of trials, when to stop rolling?

Suppose you have one $k$-sided die (labeled $1, \ldots, k$). You want to roll the highest number you can. You get $T$ trials. When you roll, you can either accept the number you rolled, or you can ...
30
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6answers
11k views

100 prisoners and a lightbulb

100 prisoners are imprisoned in solitary cells. Each cell is windowless and soundproof. There's a central living room with one light bulb; the bulb is initially off. No prisoner can see the light bulb ...
6
votes
1answer
847 views

Strategy in an urn problem with two urns

We consider two urns each containing $N$ balls. One of the urns has white balls only. The other has $k$ black balls and thus $N-k$ white balls. Both $N$ and $k$ are known, and $k>0$. What would be ...
0
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3answers
13k views

Probablity of picking up 3 vowels from the word “MATHEMATICS”

Each of the letters in the word "MATHEMATICS" is on a letter tile in a bag. Foool picks three without replacement. what is the probability that he will get all vowels? My approach, The number ...
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2answers
1k views

Is This Conditional Probability Problem asking for Pr(A|B) or Pr(A and B)?

A jar contains black and white marbles. Two marbles are chosen without replacement. The probability of selecting a black marble and then a white marble is 0.34, and the probability of selecting a ...
8
votes
2answers
3k views

On the number of consecutive tails when flipping a biased coin

Say we flip a biased coin such that the probability of getting the same outcome in a row (head-head or tail-tail) is $p$. What is the probability of getting three or more tails consecutively out of $...
0
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2answers
3k views

Expected value of product of two random variables

If we have a process defined as: $Y_t = ε_t + Y_{t-1}$ How do you determine the expected value: $E[Y_tY_{t-1}]$
3
votes
1answer
1k views

Expected time of mouse's survival (stochastic matrix)

In the following wikipedia page explaining stochastic matrices, there is an example with 5 boxes and a cat and a mouse where they jump to a left or right box at every turn and it explains how to ...
1
vote
2answers
208 views

Probability problem:marking as distinct

I know that such questions may be discouraged (they are on physics.SE), feel free to close. If we have 11 students and 8 identical chocolates, find the probability that a particular student gets 5 ...
3
votes
1answer
6k views

probability of three random points inside a circle forming a right angle triangle

three points are randomly chosen on a circle. what the probability that 1.triangle formed is right angled triangle. 2.triangle formed is acute angled triangle. 3.triangle formed is obtuse angled ...
3
votes
0answers
140 views

Bound from inequality

Please help me with this question. It is simple, but I am confused right now. For $i=1,...,n$, $\alpha_i \in C$ and $r_i$ independent and is such that $P(r_i=1)=1-P(r_i=-1)$. I know that $E|\sum_{i=...
2
votes
1answer
1k views

Determining probability that blue is part of an outfit based on five shirts and four ties along with a constraint for the possibilities

This problem is from Problem Solving Strategies - Crossing the River with Dogs and Other Mathematical Adventures by Ken Johnson and Ted Herr. I first let capital letters denote shirts, and lowercase ...
2
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0answers
208 views

The expectation after Bayesian inference of a Normal r.v

I'm confusing myself with this question. Suppose there is a Normal r.v $X \sim \mathcal{N}(\mu, \sigma^2)$. We known the variance $\sigma^2$ however don't know the mean $\mu$, and choose to use ...
3
votes
1answer
105 views

distribution random varible $N_K$

Suppose that $X_1,X_2,\ldots$ are sequence of independent random variables with distribution Bernoulli($p$). if $ S_n=X_1+X_2+\ldots+X_k,N_k=\min\{n\ge 1,S_n=k \}.$ find distribution random varible $...
3
votes
2answers
290 views

mathematical expectation for conditional probability

If the random variable $X$ can have positive integer values and $K=0,1,2,\ldots$ and $$P(X>k+1|X>k)=\left(\frac{k+1}{k+2}\right)^2$$ find $\mathbb{E}(X).$
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2answers
98 views

Interpretation of condition on positive random variable

Let $W$ be a random variable such that $\mathbb{P}(W > 0) = 1$ and $\mathbb{E}(W) = 1$. Is there an interpretation or motivation for the condition $$\mathbb{E}(W \log (W) ) < c$$ where $c \in (0,...
3
votes
2answers
4k views

How to calculate an expected value of maximum random variable function

Let $X \sim U(0,1)$ $Y=\max(X,0.5)$ $Z=\max(X-0.5,0)$ $W=\max(0.5-X,0)$ ask how to calculate $E(Y)$, $E(Z)$, $E(W)$
31
votes
3answers
110k views

Determining variance from sum of two random correlated variables

I understand that the variance of the sum of two independent normally distributed random variables is the sum of the variances, but how does this change when the two random variables are correlated?
0
votes
2answers
852 views

Random Walk on the 1D Lattice With Barriers

The walk on the 1D Lattice with no barriers can be illustrated where a walker starts at zero (0) on the number line and a fair coin is flipped. If it lands on heads, the walker moves one (1) unit to ...
2
votes
3answers
58k views

How to solve Probability questions? [closed]

I am preparing GRE Test which has Math Part. How to solve probability questions in a real quick ? One marble is randomly selected from a bag that contains only 4 black marbles, 3 red marbles, 5 ...
2
votes
2answers
376 views

Statistics resources with examples for a C.S. student

I'm a computer science student and is fairly familiar with basic probability (calculating the probability of a event occurring, pmfs and pdfs) but I find it very difficult to grasp the concepts of ...
5
votes
1answer
289 views

Stochastic Integral which is almost surely zero at fixed time

This is an exercise from Karatzas and Shreve. Find a $(Y_s)_{s \in [0,1]}$ progressively measurable such that $ 0 < \int_0^1 Y_s ^2 ds < \infty$ almost surely, and $\int _0^1 Y_s dW_s = 0$ ...
2
votes
1answer
4k views

dice throwing probability. Get face 1 at least one time by throwing a die 6 times.

How can I calculate the following probability: Throwing a die 6 times, what is the probability of having face no. 1 showing at least one time.
0
votes
1answer
810 views

Probability question using binomial coefficient

I am trying to solve the following problem : A biased coin which has P(heads) = p = .7 and p(tails) = q = .3 is tossed 3 times. the coin is tossed in such a way that the outcomes on each toss are ...
0
votes
2answers
2k views

is P(A|B) the same as P(A/B)?

part of a question that I am trying to asks to "use the definition of conditional probability along with results of parts (b) and (c) to calculate P(has disease/positive test)". Now I am confused as ...
2
votes
2answers
72k views

a probability question using percentages

This question is confusing me as I am not used to seeing percentages in a possibility question. ...
2
votes
1answer
17k views

conditional probability of an ace being drawn and then a king

I am trying to study for my statistic exam and not sure how to solve this question. The question is : what is the possibility of the following event: an ace is drawn first and a king is drawn ...
1
vote
2answers
13k views

Fair four sided die is rolled twice, what is the possibility of the sum to be 4 or less?

A fair four sided die is rolled twice, what is the possibility of the sum of the 2 die rolls to be 4 or less? could you please explain in detail
1
vote
1answer
388 views

How to calculate Non-integer raw moment of Beta random variable?

If $B \sim B(1, \beta) $ is a beta random variable. We know that its $k$ raw moment is $$ E(B^k) = {\binom {\beta + k} k}^{-1} $$ But how can we calculate $E(B^k)$ when $k > 0$ is not an ...
1
vote
1answer
72 views

How should I generate this random variable?

Suppose we now have a Erlang distribution $b(x;2,1) = x e^{-x}$. According to the definition of Erlang distribution we know that the variance of such a distribution is 2 and we want to reduce the ...
32
votes
3answers
46k views

pdf of the difference of two exponentially distributed random variables

Suppose we have $v$ and $u$, both are independent and exponentially distributed random variables with parameters $\mu$ and $\lambda$, respectively. How can we calculate the pdf of $v-u$?
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1answer
288 views

Probability and Odds when Gambling!

Here is a 'lunch break' problem from a rather old publication. Devise one set of rules for a dice game, where any number of players and one representative of the bank (mandatory), with one die each,...
3
votes
1answer
205 views

Estimate the sample deviation in one pass

We've learned this algorithm in class but I'm not sure I've fully understood the correctness of this approach. It is known as Welford's algorithm for the sum of squares, as described in: Welford, B.P....
-1
votes
1answer
108 views

calculating the expectation

Let $Y$ and $X$ be independent centered normal random variables with variances $\frac{\sigma^2}{1-\theta^2}$ and 1 respectively. How can I compute $$E\left[\frac{YX}{Y^2 + (\theta Y+X)^2}\right]$$ ...
2
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2answers
1k views

How to find this probability?

In a company, it's known that $45 \%$ of the employees have a graduate (university), a half are not graduate and make less than $\$1500$ per month.Those who are graduate, $\frac{2}{9}$ make less than $...
2
votes
3answers
541 views

Prove that $N$ has a Poisson distribution with parameter $\lambda$.

Let $\lambda > 0$, define the random variable $N$ as follows: where $U_1$, $U_2$, $\ldots$ is a sequence of iid $U(0,1)$ random variables. $U_1 \geq e^{-\lambda}, U_1U_2 \geq e^{-\lambda}, U_1U_2 \...
1
vote
2answers
78 views

How to show that these two random number generating methods are equivalent?

Let $U$, $U_1$ and $U_2$ be independent uniform random numbers between 0 and 1. Can we show that generating random number $X$ by $X = \sqrt{U}$ and $X = \max(U_1,U_2)$ are equivalent?
0
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2answers
126 views

Finding a distribution of a random variable generated using a Monte Carlo method

I would greatly appreciate if somebody could confirm or negate my result to the following problem. I am especially not sure about "putting it all together" step. Generate $U_1,U_2,U_3 \sim U[0,1]...
0
votes
1answer
616 views

How to easily find limits in transformation problems

If X and Y has joint pdf, f(x,y) =1 0 < x < 1, 0 < y < 1, and we want to find the pdf of Z = X +Y what is an easy way to do this? The hard part about this problem is determining the limits....
1
vote
2answers
223 views

Second-order Bayesian inference

We have three events $A$, $B$ and $C$ in question, and given appropriate priors, we derive the posterior $\Pr(A|B)$. Now we want to derive a 'second-order' posterior $\Pr(A|B,C)$ by using the 'first-...