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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14 views

Numerical CDF of sum of dependent lognormal variables

Let $\left\lbrace x_1,x_2 \right\rbrace $ be a general bi-variate dependent lognormal distribution, with means $[\mu_1,\mu_2]^T$ and covariance matrix $\Sigma = \begin{pmatrix} \sigma_1^2 & ...
1
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1answer
14 views

How to prove that doing something earlier in time increases the probability of success?

Let's say I have two events A and B. If I do A before B I will succeed. If I do A after B I will fail. Now, the problem is that I don't know when B happens. Intuitively I can say that if do A as ...
0
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0answers
11 views

What is the probability that in an estate of 10,000 computers ( PCs, laptops etc) that work optimally for up to 4 years?

I have 10,000 computers ( PCs,laptops etc) in my organsiation. I only have the funding to replace 500 computers a year ( this varies year on year depending on capital allocations). If I am only able ...
1
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1answer
223 views

One-tailed two-sample T-test OK?

I'm trying to conduct a one-sided hypothesis test between two random variables which are both asymptotically normally distributed with different variances. The variances are not known but have been ...
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0answers
24 views

Birthday Problem

This is an extension of birthday problem, please help In a class of 85 students, let X be the number of students who share a birthday with at least two other members of the class. a) ...
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2answers
22 views

Geometric distribution related probability questions [on hold]

I am learning Probability, and I have this problem. Suppose $X\sim {\cal Geom}(p_1)$ on $\{1,2,3,...\}$, $Y\sim {\cal Geom}(p_2)$ on $\{1,2,3,...\}$, and $X, Y$ are independent. Let $S=X+Y$. ...
0
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1answer
25 views

Conditional expectation of function of random variable [on hold]

Show E[u(Y)|Y]=u(Y) given that u(Y) is a real function.
2
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2answers
46 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: ...
0
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0answers
9 views

Inference Two Sample

During the regular season of the NBA, the great Larry Bird made 3,960 free throws out of 4,471 attempts for a career free throw percentage of 88.6%. As of March 21/11, British Columbia’s own Steve ...
0
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1answer
14 views

Probability with two uniformly distributed cost

I have two uniformly distributed costs that are random and statistically independent. They are 3,000 to 6,000 and 3,000 to 12,000. I am trying to find the probability that the total cost of these two ...
0
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2answers
15 views

Variance of a dice roll

I am currently working on a problem and am unsure if I approached it correctly. Here it is: ...
3
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1answer
606 views

Expected number of times random substring occurs inside of larger random string

I have a four-letter alphabet containing A, B, C, and D. What is the expected number of times a string of length $m$ occurs inside of larger random substring of length $n$, both generated from the ...
14
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1answer
220 views

The problem of the most visited point.

Represents the set $R_{n\times n}=\{1,2,\ldots, n\}\times\{1,2,\ldots, n\} $ as a rectangle of $n$ by $n$ as points in the figures below for exemples. How to calculate the number of circuits that ...
0
votes
1answer
18 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 ...
0
votes
1answer
29 views

Transformation theorem, Cauchy distribution

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

Determining level of significance when hypothesis is an interval

For a normally distributed sample: σ = 60 Sample size = n = 12 Sample mean = x = 3450 Null Hypothesis = H_o ≠ 3500 Hypothesis = H_1 = 3500 I need to determine the smallest significance level at ...
2
votes
3answers
43 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 ...
3
votes
2answers
25 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. ...
0
votes
1answer
14 views

how to find the maximum of the cross-entropy of a discrete random variable?

For a discrete random variable $x$, the cross entropy is $$H(x) = -(p_1\log p_1+\cdots+p_n\log p_n)$$ , so what is the maximum of $H(x)$? Here is what I tried, I compute the gradient as follows ...
0
votes
1answer
27 views

Using joint probability density function to find the conditional probability of an event [on hold]

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
votes
2answers
39 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
153 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
28 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 $[0;1]$. ...
30
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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 ...
0
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0answers
37 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
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1answer
21 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. ...
1
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1answer
62 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
26 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
28 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
15 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
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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, ...
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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 ...
1
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1answer
49 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 ...
1
vote
2answers
213 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
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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
21 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
42 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 ...
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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
43 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
50 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
30 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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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
39 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
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1answer
45 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 ...