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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1answer
21 views

Using joint probability density function to find the conditional probability of an event

Obtain $P(2<Y<3 | X =1 )$ where the joint pdf of $X$ and $Y$ is $f(x,y) = (6-x-y)/8$ where $0 < x < 2$ and $2 < y < 4$. I have no idea about this question. can someone solve it for ...
0
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0answers
12 views

Transformation theorem, Cauchy distribution

I have derived the density for the ratio of two independent random variables,via the transformation formula: $V = X/Y $ and $ U = X $ inversion yields: $Y = U/V$ och $X =U$ , the jacobian: ...
0
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2answers
27 views

Probability notation P versus Pr

I have come across both $P(…)$ and $Pr(…)$ being used to represent probabilities. Is there any difference in the meaning of these notations, or are they just different shorthands? I seem to come by ...
5
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2answers
152 views

50/50 Joker of “Who wants to be a Millionaire” - A “Monty Hall Problem” variation?

So the Monty Hall Problem itself is widely known and understood. Nonetheless, a friend of mine and I were wondering whether the the same strategy could affectively be applied by a participant of ...
2
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0answers
23 views

How to scale “probabilities” to a given mean?

I have a set of scores $x_i$, $i=1,\ldots,N$ (mimicking probabilities, $0\le x_i\le 1$) and I want to transform them so that the result has a given mean $m$, while remaining in the interval $[0;1]$. ...
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0answers
13 views

Calculating probability of a time-series probability crossing a threshold

(Please feel free to suggest a better title -- I'm still not sure what to call this in the first place.) I'm having trouble getting my head wrapped around a time-series stochastics problem I've run ...
29
votes
3answers
5k views

Making 400k random choices from 400k samples seems to always end up with 63% distinct choices, why?

I have a very simple simulation program, the sequence is: Create an array of 400k elements Use a PRNG to pick an index, and mark the element (repeat 400k times) Count number of marked elements. An ...
2
votes
2answers
36 views

Easier way to solve conditional probability question?

Two digits are chosen at random from a table of random numbers containing the digits 0,1,2,...,9. Find the probability that the second number is 2, given that the sum is of the two numbers is greater ...
0
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0answers
34 views

What did I do wrong when using Jacobian transformation

A device containing two key components fails when, and only when, both components fail. The lifetimes, $T_1$ and $T_2$, of these components are independent with common density function $f (t) = ...
2
votes
1answer
17 views

Does “Expected Absolute Deviation” or “Expected Absolute Deviation Range” exist in stats and have another name?

So everyone is familiar with Variance and Standard Deviation from high school, but it seems no one has any familiarity with a philosophical justification for such weird, seemingly arbitrary measures. ...
3
votes
1answer
19 views

correspondence between balls in compartments and integer vectors

I'm doing a self-review of probability and working through Ross' Introduction to Probability. The question is (Ross, ch2 number 51): suppose $n$ balls are randomly distributed into $N$ compartments. ...
1
vote
1answer
55 views

Obtaining PMF from a binomial distribution (Joint)

$$X\sim\mathrm{binomial}(1, 1/3)\text{ and }Y\sim\mathrm{binomial}(2,1/2)$$ How can I get $$W = XY+1$$ Normally I would attempt but this one I don't even know how to get started
0
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0answers
20 views

Can someone help me with the following math question/dilemma?

I have a pool of objects that are randomly selected from a global object database. The objects certain numeric attributes: The objects from the pool are fed to users in real time Users will either ...
1
vote
1answer
25 views

How to show that $\Phi(1-x)^{-1} =O(\sqrt{\log{x^{-1}}})$

In the middle of some proof, I have faced an expression $\Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}})$, where $\Phi(\cdot)^{-1}$ is a quantile function of the standard normal distribution and $x \in (0,1)$. ...
1
vote
1answer
11 views

Find the probability $P[ x(t) \le 1]$ where $x(t)$ is a filtered Poisson process (rect pulses)

I can't understand the following question: "The random process x(t) is defined as $$x(t) = \sum_{n=- \infty}^{+\infty} rect(\frac{t-\tau_{n}}{T}) \quad ,\quad t \ \epsilon \ (R)$$ where {$\tau_{n}$} ...
2
votes
2answers
332 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 ...
0
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0answers
60 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
0
votes
1answer
23 views

Finding $V(X)$ when you don't have a density/distribution function.

I just did the first part of this problem: You have a lot of $50$ items and are taking a sample size of $15$. In the lot $3$ items are defective. The lot is accepted if the number of defective items, ...
-1
votes
1answer
330 views

how many elements are in A? (sets)

Five applicants (Jim, Don, Mary, Sue, and Nancy) are available for two identical jobs. A supervisor selects two applicants to fill these jobs. Let A denote the set of selections containing at least ...
0
votes
1answer
59 views

Probability and coin tosses

Taking a Probability & Statistics class this term and trying to get my head wrapped around how I calculate coin tosses with specific out comes in mind. We're using the nCr and nPr functions on our ...
0
votes
1answer
41 views

Expected number of changes of serves in a game of raquetball

Suppose a game of racquetball is being played, with players A and B. Assume further that A starts the play first, that the winner of a point serves the next point, and that the match lasts until the ...
2
votes
0answers
32 views

Conditional expectation of the sum of two random variables

I've got some difficulties in calculating the conditional expectation of the sum of two RV. I am not sure if I correctly formalized the scenario I am looking at. So I am trying to describe it first: ...
1
vote
2answers
211 views

Relationship between expected and actual value

Consider a zoo with two animals: an elephant and a lion. The elephant's expected lifespan is $70$ years, but it lives to be $65$ years ($SD = 7$). Likewise, the lion's expected lifespan is $22$ years, ...
0
votes
2answers
45 views

Expectation of CDF of continuous random variable $X$, evaluated at $X$

Given the continuous random variable $X$ with cumulative distribution function $F_{X}$, find $E[F_{X}(X)]$. Attempt at solution: I understand that the expected value, $E[X]$, of a random ...
2
votes
1answer
20 views

The probability of getting a certain image by random pixelation

Well, seeing that I'm terribly bad at math I don't know how to solve this, I'll try to explain, excuse me if I sound dumb. Just suppose that I've got a photo/image with 320x240 resolution and 24 bit ...
0
votes
1answer
38 views

Probability of two teams meet up in FIFA tounament

**Second round is single elimination round. **Tournament is from 16 teams elimination follower by quarter finals,semi-finals and a final. **The losing semi-finalist contest a third place match. In ...
5
votes
1answer
186 views

Asymptotics of sum of binomial distributions

Definition 1: For any random variable $X$, we define $\mathrm{Bin}(p,X)$ as a variable with binomial distribution having parameters $p$ and $X$. Definition 2: For all $i \in \mathbb{N}$, define ...
-5
votes
0answers
35 views

what is the probability of getting two heads twice in 5 tosses of two coins? [on hold]

If I have two identical fair coins and toss them both at the same time, what is the probability that after five tosses, two tosses resulted in both coins landing heads up?
2
votes
1answer
41 views

Expectation related to Normal distribution and its density

Given $\sigma^2>0$. Let $Z\sim N(0,1)$ and $\Phi$ be the cumulative standard normal with density function $\phi$. I wish to show that $$ E\left(\frac{Z^2}{[\phi(\sigma Z)]^2}\Phi(\sigma ...
1
vote
1answer
49 views

Calculate single “battle” outcome odds for RISK

I am trying to reproduce the values in this odds ratio table from Wikipedia. For all those unfamiliar with RISK, this is a game where units fight against each other via the roll of the dice: The ...
0
votes
2answers
29 views

probability of a flipped coin

A fair coin is flipped three times. Let $A$ be the event that a head occurs in the first flip and $B$ be the event that exactly one head occurs. a) Find $p(A/B)$ b) Are $A$ and $B$ independent? ...
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votes
2answers
30 views

If pages in a book have an iid Poisson number of errors, in 10 pages what is the probability that exactly 3 pages have exactly 1 error?

Suppose the number of spelling error on any given page in particular book can be modeled by a Poisson distribution with $\lambda=2$, and assume that the number of errors on different pages is ...
6
votes
3answers
157 views

Is there a simple way to illustrate that Fermat's Last Theorem is plausible?

A first step in proving a theorem is true could be to show that it is plausible, so at least you then would have a general idea that it could be true and have something to start with in proving it. ...
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0answers
38 views

Choosing random marbles until one is divisible by $X$ [on hold]

A box contains twelve marbles on which they are numbered by $1,2,3,...,12$. Now let $X$ represent the number of marbles you must choose with replacement until you obtain one with a number that is ...
2
votes
2answers
51 views

conditional probability about sum and product rule

I am reading Bishop's Pattern Recognition and Machine Learning. In page 73, chapter 2.1. I can't understand the formula 2.19 : $$p(x=1|\mathcal{D})=\int_0^1 p(x=1|\mu)p(\mu|\mathcal{D})\text{d}\mu ...
1
vote
1answer
42 views

Probability of getting a right answer?

A student is taking a $4$ question multiple choice quiz with each question having $5$ options. What is the probability that he will get at least one question correct? P.S. Please keep answers at ...
2
votes
1answer
37 views

A question about Malliavin calculus

An application of Malliavin calculus is to calculate the sensitivity of financial Greeks. However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to ...
0
votes
2answers
26 views

Convergence of running maximum of uniform random variables [on hold]

Let $X_1, X_2, ... X_n$ be an IID sequence of IID random variables that have a uniform distribution $(0,1)$. Let Max$(n) =$ max$(X_k:1\le k \le n)$, where $n\in \mathbb N$. How do I show that ...
0
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2answers
28 views

conditional probability maybe?

If in application A, 70% of the users are men and 30% women. In application B, 60% men and 40% women. Given you have both applications, what is the probability that you are a man?
1
vote
2answers
37 views

Confusion regarding the fixed point $p(x) =x$

Consider an empty urn. Now at each time, we are adding one ball to it, Either red or black, the probability of a red ball being added depends on $x$ ($x$ denotes the current fraction of red balls in ...
1
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2answers
35 views

what is the distribution of a uniform r.v. divided by an exponential r.v.?

Say $U=\frac{X}{Y}$. X and Y are independent with each other. X is a Uniform distribution r.v. $X\sim \mathcal{U}(0,1)$. Y is an exponential distribution r.v., $Y\sim\mathcal{Exp}(\lambda)$, whose pdf ...
0
votes
2answers
49 views

Expected value and variance of max{x, y}

I've run into this problem while playing a game called Europa Universalis 4. I've done similar maths before in my studies so I'm pretty sure this should have an easy answer but I can't for the life of ...
1
vote
0answers
23 views

change some element of a correlation matrix

I am working on correlation matrices. These matrices have the main property to be symetric , positive-semidefinite, have 1 on the diagonal and each of their elements is between -1 and 1. Let's say I ...
0
votes
1answer
49 views

Fast way to inverse B'CB+D

$\mathbf {A = B'CB}$, where $\mathbf A$ is of dimension $n \times n$, $\mathbf C$ is m by m, positive definite and symmetric, $\mathbf B$ is of dimension $m \times n$, and $n >> m$. Inversion ...
0
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0answers
15 views

Stopped strong Markov process again strong Markov?

Following setting: I have a right-continuous strong Markov process X in a right-continuous filtration >$\mathbb{F}=(F_t)$ and a P-a.s. finite stopping time $\tau$. My question is: Is the ...
1
vote
1answer
15 views

Convex and Concave Functions using Known Function Values

I am reading the classic Prospect Theory: An Analysis of Decision Under Risk (1979, Econometrica) by Kahneman and Tversky. I am not clear on something on page 278: ...
1
vote
1answer
242 views

joint probability distribution of one discrete, one continuous random variable

This is a problem on the joint distribution of a discrete and a continuous random variable. Kitty Oil Co. has decided to drill for oil in 10 different locations; the cost of drilling at each ...
2
votes
2answers
41 views

Estimate the number of ants in a colony

A friend of mine gave me this weird problem I cannot solve. To estimate the number of ants in a colony an entomologist draws 5500 ants randomly from the colony, labels them with a radioactive isotope ...
2
votes
1answer
95 views
+50

Deducing an optimal gambling strategy (using martingales).

Apologies in advance for the length, I tried being precise. Suppose a game where in each turn you can gamble a certain amount of money on the result of a fair coin toss. If the coin comes out tails ...
3
votes
3answers
57 views

Probability of a year which is not a leap year

If a 4 digit year is choosen randomly, what is the probability that it is not a leap year ? This problem has come in my exam and i have written like this I know that the number of four digit year ...