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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4 views

Picking Marbles and Estimating the Expected Cost

This is a general version of a game I play. Suppose there is a bag of marbles that consists of 4 different marbles : Silver ...
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0answers
9 views

Probability of Level Crossing

I am kind of stuck on how to proceed on this. $X_n$ is an IID process with $$f_{X_n}(y)= \frac\lambda2 e^{-\lambda |y|}$$ There is a stationary autoregressive process $Y_n$ defined as $$Y_n=\rho ...
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2answers
44 views

Ways of coloring the $7\times1$ grid (with three colors)

Hints only please! A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the ...
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1answer
29 views

Prove that if events $A,B$ independent of C then $P(A\cap B\cap C)= P(A\cap B)P(C)$

I am trying to prove why the intersection of two events $A, B$ that are independent of C is also independent of C so that the following equality holds: $$P(A\cap B\cap C)= P(A\cap B)P(C)$$ ...
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1answer
12 views

What am I plugging in wrong to my normal distribution calculator?

I am trying to find the probability of the following question: Cans of regular Coke are labeled as containing 12 oz. Statistics students weighed the contents of 7 randomly chosen cans, and found the ...
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1answer
13 views

Discrete-time Markov Chains

I am having trouble understanding this proof from Markov Chains by Norris (1997) How do we get the equality $P_j(X_n=j \text{ for infinitely many } n ) =P_j(X_n=j \text { for some } n \ge m+1)$ ?
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1answer
29 views

Is there any difference between statistical learning and machine learning?

Straight to the point, I'm a math student and I have a course this year called Statistical Learning. From the description, the course contains: Large datasets analysis, regression, principal ...
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1answer
15 views

Expected Shortfall alternative definition

Define: $$q_\alpha(F_L)=F^{\leftarrow}(\alpha)=\inf\lbrace{x\in \mathbb{R}\mid F_L(x)\geq \alpha\rbrace}=VaR_\alpha(L)$$ I want to prove that: $$ES_\alpha = ...
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1answer
22 views

Distribution Technique Question of two independent Exponential Distributions

If $X_1$ and $X_2$ are two independent random variables having exponential densities then $f(x_1,x_2)$ is defined as $$f(x_1,x_2)=\exp(-(x_1+x_2))\,{\bf 1}_{(0,\infty)}(x_1){\bf ...
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0answers
28 views

Regression in Statistics [on hold]

A survey was carried out by a lecturer on a small random sample of her students, in which she asked them individually how much time, $y$, they spent studying and how much time, $x$, they watched ...
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0answers
21 views

the joint probability density

The joint probability density f(x, y) of a pair of random variables X and Y is zero everywhere except on and inside the L- shaped region shown in the following plot, in which the density is some ...
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1answer
31 views

Probability, step function [on hold]

A one-dimensional random walk starts at the origin x = 0 at time t = 0, and takes a random step of either +1 or -1, every second. It takes a total of 4 random steps altogether, with the final location ...
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0answers
16 views

Weighting the data by the history

I have a input stream 3D data that comes every time frame. Each point is defined by 3D vector of x,y,z. There is a evaluation function [say f(x)] that computes if the point at time t is valid or ...
2
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1answer
36 views

How many 3 letter words can you form from 'EEAAP' [duplicate]

How many 3 letter words can you form from 'EEAAP' I think the answer is ${3\choose 3} * 3! + {2\choose 1} * 3 + {2\choose 1} *3=18$. Is this right? ${3\choose 3} * 3!$ = You pick all ...
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2answers
37 views

Runs of white balls in sampling without replacement

There are $m$ white balls and $n$ black balls in a box. Balls are randomly drawn from the box with no return. Denote $X_1$ : number of white balls that been drawn before the first black. For $2 \leq i ...
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0answers
38 views

How many ways can you choose team of 5 people out of 7 men and 6 women in which there are at least 3 men?

I am confused by this question. I solved it by selecting 3 men first out of 7 men and then selecting 2 people out of 10 remaining person ( 4 men and 6 women ) . So my answer is C(7,3) * C(10,2) = ...
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1answer
27 views

Probability of the card following first ace being ace of spades or two of clubs

I am learning probability from Scheldon Ross' book. The question reads like this: A deck of 52 playing cards is shuffled, and the cards are turned up one at a time until the first ace appears. Is ...
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1answer
33 views

How do I go about solving this probability question?

I was playing around with dice this morning and flipping them, when a problem suddenly hit me. If I roll $n$ normal six-sided dice, and flip every single dice, what is the probability that the ...
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2answers
35 views

Probability and Combinatorics

I am trying to solve example 4.15 here but think the total number of outcomes in the solution is incorrect. This is my reasoning. We have 3 that qualify as best three, say BBB, and 2 as bad say OO. ...
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0answers
15 views

Quantum probability and quantum measure theory

Do quantum probability and free probability mean the same thing - that is, they deal with noncommutative random variables? What about quantum measure theory? Is quantum measure theory the foundation ...
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1answer
23 views

determination of probability

Recently I have faced a problem related to probability.I have tried it by bayseian theorem but failed.Here is the problem: A doctor knows that pneumonia causes a fever 95% of the time. She knows that ...
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0answers
21 views

A Feynman-Kac style derivation of a survival probability of a Compound Poisson process

Let $$R_t = u + \beta t - \sum^{N_t}_{i=1}U_i$$where $u\geq 0$, $\beta > 0$, $N_t$ is a Poisson counting process with intensity $\lambda$ and $U_i$ are jumps having a probability density $\nu(y)$ ...
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2answers
32 views

What is the variance of the volumes of particles?

According to Zimmels (1983), the sizes of particles used in sedimentation experiments often have a uniform distribution. In sedimentation involving mixtures of particles of various sizes, the larger ...
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2answers
30 views

Class Coin Toss Experiment

My classmates and I are doing a coin toss experiment (i.e. toss coin 100 times). I have already determined that I have a fair coin, since I tossed $43$ heads, and this falls into a $95$% confidence ...
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1answer
30 views

Expected value and single event probability [on hold]

Can expected values be added together to represent the overall expected value of a single event? Or is there a different way to do expected value for a group of probabilities? Also if I have 10 bets ...
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0answers
21 views

What is the proof behind the mean confidence interval for a Binomial Distribution?

How do we obtain the range to be as [$\mu-$$zσ$, $\mu+$$zσ$]? Is it when $n$ is sufficiently big?
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0answers
29 views

Probability of a run of *k* or more of a subset of categories in *m* multinoulli trials?

Given a multinoulli distribution of categories $(C_1,C_2,...,C_n)$ with associated probabilities $\left\{p_1,p_2,\ldots ,p_n\right\}$ with $\sum _{i=1}^n p_i=1$, is there a tractable way to get the ...
-1
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0answers
31 views

Odds for rolling specific faces on a 3-sided die [on hold]

Firstly, thank you for taking my question! Imagine five (5) $3$-sided dice, so three unique faces. What are the odds (percentage) of rolling $3$ of the same face when rolling $5$ dice? When rolling ...
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1answer
26 views

lottery question [on hold]

for the lottery -- if I have 4 numbers How can I see all of the 4 number combinations, never using the same number more than once in each combination
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2answers
49 views

100-sided dice was rolled 98 times, how do you choose next numbes to bet, based on current outcomes.

100-sided dice was rolled 98 times, Numbers form 1 to 50 were rolled exactly once, except number 25, which wasn't rolled yet. Number 75 was rolled 49 times You can only bet if the next roll result ...
2
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3answers
277 views

Is the limit of càdlàg functions càdlàg?

Is the pointwise limit of càdlàg functions càdlàg? If not which are the weaker conditions to assure it? I cannot find a counterexample
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1answer
21 views

Is there an upper bound for expectation of product of two measurable function on a random variable?

I wonder if there is an useful upper bound for $\mathbb{E}_{x\sim p(x)}[f(x)g(x)]$ in the following form: $$ \mathbb{E}_{x\sim p(x)}[f(x)g(x)] \leq \mathbb{E}_{x\sim p(x)}[f(x)]\times xxxxxx $$ The ...
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0answers
48 views

Probability - There is a radar, a computer and a gyroscope [on hold]

There is a radar, a computer and a gyroscope on board an airplane. The probability that the radar fails is 0.2. If the radar fails, the gyroscope will also fail, and the probability that the computer ...
3
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2answers
77 views

Combinatoric Birthday Paradox

There is likely a closed form solution for this problem but it's had me puzzled for days. This is about a variant on the classic birthday paradox. To recap, the birthday paradox is where given only 23 ...
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1answer
15 views

Distribution of specific distributions

I have a normal distribution of independent variables, and there are a specific number of samples to this distribution: say 1 million samples. A function is set by the largest value of these million ...
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0answers
21 views

Scrabble/words with friends [on hold]

How many letter combinations are possible with 7 tiles? Just the math answer please, 7 tiles in 7 slots, how many different combinations? Thank you :)
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2answers
28 views

Why is this counting function finite? (It is used Probability)

Why is this counting function finite? I don't understand this interpretation of the author. Can you explain more about this? Please.
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0answers
24 views

Distribution of the test statistic?

Let $\mathbf{x}_i \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma)$. I am trying to find a distribution of the following test statistic $ T(\mathbf{x}) = \frac{\bar{\mathbf{x}}^H ...
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0answers
13 views

Meaningful Extreme value distribution

Extreme value theory (EVT) dictates that the limit distribution of the minimum of the set of i.i.d. Chi-square random varibales $\{C_1,C_2,\cdots,C_n\}$ is Weibull. The Weibull distribution has ...
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3answers
34 views

Probability of picking a card one out of 52 times.

Let's say we have a standard deck of 52 cards. What would be the probability of choosing the 2 of diamonds? Obviously, it would be $\frac{1}{52}$. If we were to randomly choose another card from ...
0
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1answer
30 views

Probability that one normal Random Variable will fall within a given range of another.

I'm struggling with the following problem: (ed: Don't be lazy. Just type it out. ) A certain small freight elevator has a max. capacity $C$, which is Normally distributed, with mean ...
1
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1answer
26 views

Probability of 2 students being chosen the both have under 100 books at home

Suppose we select two students at random from the class of fifteen. What is the probability that both students chosen have less then 100 books at home? Data provided is the amount of books each ...
1
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1answer
14 views

Relationship between minimizing a conditional variance and a covariance

We are working with discrete-time stochastic processes. Let $v_k$ be a $\mathcal F_k$-predictable process, and let $X_k, \eta_k$ be $\mathcal F_k$-adapted processes. Define $V_k = v_kX_k+\eta_k$ and ...
-2
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2answers
35 views

How to solve this probability formulation? [on hold]

$\int_{200}^{250} P(a=x \land 450-x \leq b \leq 250)\space dx$, where $a$ and $b$ are uniformly distributed random variables on $(0,250]$ and $(10, 250]$ respectively.
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1answer
50 views

Probability question from GRE subject test

I ran across the following problem while practicing for the GRE math subject test: Suppose $X$ is a discrete random variable on the set of positive integers such that for each positive integer $n$, ...
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0answers
33 views

How to calculate $P(X_1 < X_2 < X_3…X_n ) $ [on hold]

Could you please help with the following problem i am having- I need to calculate the probability of $X_1$ (randomly selected discrete value between $a$ and $b$) being smaller then $X_2$ (randomly ...
1
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2answers
37 views

Find the following probability

A bowl contains 16 chips, of which 6 are red, 7 are white and 3 are blue. If four chips are taken at random and without replacement, find the probability that there is at least 1 chip of each colour. ...
2
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2answers
54 views

Probability distribution of number of waiting customers in front of a counter

The number of customers arriving at a bank counter is in accordance with a Poisson distribution with mean rate of 5 customers in 3 minutes. Service time at the counter follows exponential distribution ...
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0answers
31 views

Show that the following set function is not a probability set function

If the sample space is $\mathfrak{C} = \{c : -\infty < c < \infty\}$ and if $C \subset \mathfrak{C}$ is a set for which the integral $\int\limits_C e^{-|x|}dx$ exists, show that this set ...
2
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2answers
27 views

Distribution of a product of Multinomials

Consider the following: $(X_1, X_2, X_3, X_4) \sim \mathrm{Multinomial} (n,\mathbf{p})$ where $\mathbf{p} = (p_1,p_2,p_3,p_4)$. I would like to find the distribution of $X_1 X_4$, or at least know ...