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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Probability & Statistics

You and a friend play a game in which the winner is the first player who has 7 or more points and is 2 points ahead of the other player. Note that game involves rounds of play, and the winner gains ...
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2answers
87 views

Need help in question for conditional expectation

Let $(X, \Sigma, P)$ be the probability space, let $X$ and $Y$ be a integrable random variables (all are bounded) on this space and $\Sigma_{0}$ be a sub-$\sigma$-algebra of $\Sigma$. Show that $$ ...
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143 views

Covariance and Correlation

Suppose there were m married couples, but d of these 2m people have died. Regard the d deaths as striking the 2m people at random. Let X be the number of surviving couples. Find: a) E(X) b) Var(X) ...
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268 views

Correlation of Indicator Variables

Show that for indicator random variables $I_A$ and $I_B$ of Events $A$ and $B$: $Corr(I_A, I_B) = Corr(I_A^c, I_B^c) = -Corr(I_A, I_B^c) = -Corr(I_A^c, I_B)$ Deduce that if $A$ and $B$ are ...
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1k views

What is the probability that both the fuses are defective? [closed]

A box contains $20$ fuses of which $5$ are defective. If $2$ fuses are chosen together at random, what is the probability that both the fuses are defective?
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71 views

Conditional Probability (twice applied)

I am trying to better visualize why: $$P(A|B,C) = [P(B|A,C)P(A|C)]/P(B|C)$$ I know we can get this by doing: $$ \,\,\,(1)\, P(A|B,C) = P(A,B|C)/P(B|C) \\ (2)\,P(A,B|C) = P(B|A,C)P(A|C)$$ I can ...
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1answer
199 views

Expectation of a random variable uniformly distributed according to another random variable

Let's say $X$ is a random variable with P.D.F. $f_X(x) = 3x^2$ for $0 < x \leq 1$, and $0$ otherwise. Further, $Y$ is a random variable distributed uniformly on $[0, X]$. Find $E[Y]$ I'm ...
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139 views

Question about Conditional Expectation

I have a seemingly trivial question regarding conditional expectation. Consider $x$ and $y$ be two integrable random variables on Probability space (X, $\Sigma$, $P$) such that $$E(X|Y) =_{a.s} Y$$ ...
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2answers
128 views

Geometric probability question

So there are 2 parallel lines 20 feet apart. A piece of pipe 20 feet long falls between the lines and one end is exactly 10 feet from one line. What is the probability that the pipe lies entirely ...
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93 views

Algebra 2 Probability And Combination Problem

Here is the problem: You have a book collection that consists of 20 horror novels, 15 romance novels, and 25 mystery novels. You randomly pick 4 books to read during a long trip. What is the ...
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1answer
57 views

Probability of $X^{2} > Y^{3}$ over distribution other than uniform

I was working on a problem in which $X$ and $Y$ are continuous random variables which both have the uniform distribution $U[0, 1]$. The question was: what is the probability that $X^2 > Y^3$. ...
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1answer
39 views

Easy question about Gamma distribution

Let a Gamma distribution (gamma(n,λ)) , theoretically How the values ​​of n and λ affect to the graph? Does the gamma distribution has any particular form , or depends on the choice of parameters? ...
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4answers
188 views

What exactly is a probability measure in simple words?

Can someone explain probability measure in simple words? This term has been hunting me for my life. Today I came across Kullback-Leibler divergence. The KL divergence between probability measure P ...
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1answer
579 views

“Square root” of a normal RV?

Say $X_1,X_2$ are independently drawn from the same distribution (call it $X$) and that their product, $X_1X_2$ falls on a standard normal distribution. Is it possible to get a pdf or cdf for $X$? ...
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1answer
71 views

Conditional probability and linearity

I am reading some introductory notes to probability theory and I am puzzled by the sentence that follows. After having defined conditional probability as $$\text {Pr} (A|B) = \text {Pr}(A\cap B)/\text ...
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1answer
59 views

Integration of PDF

Let $X$ and $Y$ be random variables. I need to find $Y$ conditional probability density function, when: a) $f_{xy}(x,y)=\lambda^2exp(-\lambda y)$, $0\leq x\leq y< \infty$ b) $f_{xy}(x,y)=x ...
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1answer
114 views

Conditional probability Bayes Theorem

I am trying to solve this problem but I am not sure how to obtain the formula given below. Any help would be appreciated. A boy is selected at random from among the children belonging to families ...
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1answer
197 views

Probability of two people in a group of n people sharing the *exact* birthday?

I understand the solution to the birthday paradox. But I was wondering how I would calculate the probability of two people having the same age, or the exact birthday, down to matching years. I am ...
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1answer
108 views

How far to look before line-of-sight will intersect a star

I was told this sort-of riddle by someone, having to do with the proof for the finite age of the universe, and I'm not sure how to approach the answer. Assuming that the entire universe is uniformly ...
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1answer
87 views

Picking a discrete set in a continuous probability distribution

maybe this a stupid question, however I could not solve it properly. What´s the general method to solve problems relating the probability of a given event in a set of discrete events picked from a ...
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1answer
166 views

De Moivre's martingale and stopping times

I'm reading Grimmett's and Stirzaker's book of probability and random processes, and there's an example on De Moivre's martingale which confuses me (pp. 492, if you want to read along). Consider a ...
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1answer
50 views

Permutation problem - probability bound

I have a similar question as this Random permutation problem, where X := the number of inversions in  π . But I need to find an upper bound on the probability that X is more ...
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0answers
79 views

Stopping time and expectations

Let $(X_n)_{n\geq 0}$ denote a martingale with respect to some filtration $(\mathcal{F}_n)$, and let $\tau$ denote a stopping time with $P(\tau<\infty)=1$. Assume that there exists an $M$ such that ...
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2answers
63 views

A question on characteristic functions

I am trying to understand a proof of the following: Let $X_1,...,X_n$ be stochastic variables in $\mathbb{R}^n$. Then the following statements are equivalent: i) $X_1,...,X_n$ are independent. ii) ...
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1answer
79 views

A modified Buffon's needle

A needle 2.5cm long is dropped on a piece of paper that has a very fine parallel lines 2.25cm apart drawn on it. What is the probability that the needle lies between the two lines? I can see how ...
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3answers
91 views

Which probability law?

It may be a basic probability law in another form, but I cannot figure it out. Why can we say the following: $P(A∩B|C) = $$P(A|B∩C)P(B|C)$ Thank you.
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2answers
232 views

Combinatorics/Probability Distribution Example Question

At a local fast-food restaurant in Oregon (no sales tax), fries, soda, hamburgers, cherry pie, and sundaes cost \$1 each. Chicken sandwiches cost \$2 each. You have five dollars. How many ...
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0answers
111 views

Balls and bins (the power of two choices)

Still balls into bins. Suppose there are N balls and N bins, with the power of two choices, i.e., for each ball, randomly select two bins and place the ball into the bin with least loads. What's the ...
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2answers
182 views

Other way to express $e^{|x|+|y|}$

I have a joint PDF with $e^{|x|+|y|}$. I know I can separate the function in two functions, $e^{|x|}$ and $e^{|y|}$. The limits for $x$ and $y$ are from $-\infty$ to $\infty$. Can I integrate from $0$ ...
2
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1answer
64 views

Convergence of a random variable

I'm working on a problem and I appreciate if you can guide me how to proceed. Assume that $\sqrt{n}\big(X_n - \mu\big)\rightarrow^d N(0,1) \text{ as } n\rightarrow \infty$. By the symbol ...
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1answer
499 views

Definitions for an exponential family to be curved or flat?

I was wondering how a curved exponential family is defined? Also how is a flat exponential family defined? Is "curved" or "flat" defined for a family of probability distributions, or for a ...
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2answers
651 views

Help solving part B!

A multiple choice test consists of 100 questions. Each question has 5 possible answers, only one of which is correct. Four points are awarded for each correct answer, and 1 point is taken off for each ...
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0answers
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Expectation Optimization

You have $\$1.50$ and are given the opportunity to risk any portion of the $\$1.50$. This portion you risk has a chance of one in ten to return $10$ times the value. How much should you risk if any ...
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1answer
42 views

How can I calculate the probality of to get a number with 15 digits lenght repeat

A number with nineteen digits is generated randomically by a particular system and it's guaranteed that every number generated is unique (by the system provider). If I chunk this number and get the ...
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1answer
57 views

Random sampling

This is a very simple question, but I'm having a dilemma here. Given a set (1, 2, 3, 4), find the sampling distribution of the mean S of sample of size 2. 1) Way 1: Possible output are: 1.5 2 ...
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3answers
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Probability of streaks

I thought of this question lately, but I'm not satisfied with the answer I got: If I flip a coin 100 times, what is the probability that I will get a streak of at least ten of the same side? The way ...
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1answer
77 views

Special probability distributions

In a box there are $10$ balls numbered $1,2,...,10$. We're taking out balls one by one until we take out the ball numbered $5$. Let $X$ be the number of balls take outs, find the probability ...
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1answer
130 views

Probability of choosing C out of T students with restrictions

There is a class of $T$ students, consisting of $G$ girls and $(T - G)$ boys. Out of the $T$ students, only $C$ are selected for an examination. What is the probability that there are at least $K$ ...
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1answer
542 views

Find the maximum likelihood estimator for $\theta$ when $f(x)=2\theta^{-2}x, 0\leq x \leq \theta$

Find the maximum likelihood estimator for $\theta$ when $f(x)=2\theta^{-2}x, 0\leq x \leq \theta$. This should be a really easy question but I somehow cannot seem to get the right answer. My ...
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1answer
675 views

Dice Roll Permutation Problem

Here is my problem: You have a standard dice, with possible rolls: $\{1, 2, 3, 4, 5, 6\}$. How many permutations exist in 10 rolls such that no two immediate rolls are the same? For example: $\{1, ...
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3answers
168 views

Expected number of turns

A person has $n$ keys and one lock. After every failed attempt to unlock, the person may pick a new key and discard the old one. What is the expected number of attempts needed to open the lock?
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0answers
96 views

Increase difficulty level based on probabilities

Increase difficulty level based on probabilities. Hello. I’m creating a bejeweled type of game and I’m trying to implement the difficulty of levels. My game is structured this way. The board is 10 by ...
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55 views

Probability in a game

Let there be a game of choosing exactly $B$ coins from $A$ coins. ($A$ coins include $1$ coin of player $X$, $C$ coins of player $X$'s friend and $A-1-C$ coins of player $X$'s enemy). Player ...
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3answers
100 views

Show that ${-n \choose i} = (-1)^i{n+i-1 \choose i} $

Show that ${-n \choose i} = (-1)^i{n+i-1 \choose i} $. This is a homework exercise I have to make and I just cant get started on it. The problem lies with the $-n$. Using the definition I get: $${-n ...
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3answers
232 views

Probability of a graph having at least 1 k-clique

I need to estimate the probability $P(\text{Graph G has at least 1 k-clique})$, any precision will do. I know the edge probability, say $p$, so the average number of the edges, $EK$, is $pm(m - 1)/2$, ...
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2answers
824 views

Combinatorics Distribution - Number of integer solutions Concept Explanation

I reading my textbook and I don't understand the concept of distributions or number of solutions to an equation. It's explained that this problem is 1/4 types of sampling/distributions problems. An ...
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1answer
84 views

uniform integrability characterization

How to show the following: When a family of random variables $ \{X_n\}_{n \geq 1}$ is $L^p$ bounded for some $p > 1$ then $ \{X_n\}_{n \geq 1}$ is uniformly integrable. Also why does the above ...
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2answers
65 views

pdf of a simple random variable calculated two different ways.. with two different answers

so we have a random variable Y with a uniform pdf on the interval [0,1]. the question is, what is the pdf of W=Y^2. method 1 using the transformation of variables formula: ...
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1answer
31 views

Calculate probability of choosing a winning block out of 5 blocks, extend to 9 scenarios

Suppose you have a game board with 9 rows. Each row has 5 blocks. One of out those five blocks will be a bomb. You have to choose a block in each row. In order to advance, you have to choose the block ...
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2answers
285 views

Variance for the number of trials before success in an urn problem without replacement

This question is asked as an extension of: Expectation of number of trials before success in an urn problem without replacement (Note: I am not the author of the original question.) We have $b$ blue ...