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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2answers
18 views
A basic doubt on independence of events in probability
Let $X_1$ and $X_2$ be i.i.d random variable. Now, in a book I see the following steps to calculate $P(X_1 < X_2 < x)$
$P(X_1 < X_2 < x)$
= $P(X_1 < x, X_2 < x, X_1 < X_2)$
...
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votes
2answers
22 views
A basic probability doubt on independence
Let $X_1$ and $X_2$ be two i.i.d continuous random variable. I need to find the probability that $P(X < Y)$. I know how to formally find the probability by integrating over appropriate region and ...
1
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1answer
19 views
Let $f$ be a bounded uniformly continuous function in $R^1$. Then $X_n\to 0$ in pr. implies $E(f(X_n)) \to f(0)$.
The following is problem 4 from Section 4.2 of "A Course in Probability Theory" by Kai Lai Chung.
Let $f$ be a bounded uniformly continuous function in $R^1$. Then $X_n\to 0$ in pr. implies ...
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1answer
16 views
Conditional Expectation with independent sub-sigma fields
Let X and Y be bounded random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider two independent sub-$\sigma$ fields $\mathcal{G}$ and $\mathcal{H}$ of $\mathcal{F}$. We ...
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0answers
34 views
Random Walk probability
I currently have a probability class tutorial question that I have no idea where to begin. At first instinct, I thought it may have been a CTMC question or branching question, but now I have no idea, ...
1
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3answers
37 views
Probability that a divisor of $10^{99}$ is a multiple of $10^{96}$
What is the probability that a divisor of $10^{99}$ is a multiple of $10^{96}$? How to solve this type of question. I know probability but I'm weak in number theory.
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0answers
9 views
Solving variables for the equilibrium distribution
This question is related to the equilibrium probability distribution of a markov chain. I have: $ \vec {π} * \vec {P} = \vec {π}$
$ [π_0,π_1,π_2,π_3] $ [ $\begin{matrix} 1/3 & 2/3 & 0 & ...
1
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0answers
17 views
Poisson point process convergence
Let Π be a Poisson point process on [0,∞) with intensity measure $\mu$. Assume $μ([0,t])<∞$ for all $t<∞$ and $μ([0,∞))=∞$. Also assume $μ({x})=0$ for all x. Prove ...
2
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2answers
26 views
Ties in Matching Pennies
Players A and B match pennies N times. They keep a tally of their gains and losses. After the first toss, what is the chance that at no time during the game will they be even?
I have seen a solution ...
2
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1answer
31 views
distribute m pennies to n people, what is the expectation of coins one would obtain
Assume there are $m$ pennies and $n$ people. We want to distribute the pennies to the people by uniformly picking a vector $(x_1,...,x_n)$ from the set of all vectors satisfying $x_1+...+x_n=m$, where ...
-1
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1answer
46 views
Selecting a representative permutation
If we have a complete set of permutations {m} = n choose k, how do I select a representative set of permutations in a stream such that the selected set say, {s} keeps growing to include permutations ...
2
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2answers
33 views
Probability AND/OR
Suppose we have a bag of 10 balls, and each ball is a unique colour.
If we randomly select 3 balls from this bag, without replacement, I want to find out the chances of correctly guessing the colour ...
59
votes
4answers
1k views
Probability that a stick randomly broken in five places can form a tetrahedron
Randomly break a stick in five places.
Question: What is the probability that the resulting six pieces can form a tetrahedron?
Clearly satisfying the triangle inequality on each face is a necessary ...
0
votes
2answers
22 views
A simple yet hard task for (theoretically) Poisson distribution
Sorry if I don't use the words properly, I haven't learnt these things in English, only some of the words.
Anyway, I'm practicing to one of my exams and sadly this task seemed more challanging for me ...
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1answer
28 views
Uniform distribution joint $\to$ marginal
Let vector $(X,Y)$ have a uniform distribution on the set $N = \{ (x,y): x<1,y<1,1<x+y\}$. Determine distribution $X-Y$.
So far I've thought of this:
\begin{align}
P[X | Y=y] &\sim ...
4
votes
1answer
64 views
Distribution of Digit Products
A digit product $P(n)$ of a natural number $n$ is given by the product of its decimal digits. For example:
$$P(1234) = 24,\;\;\; P(24) = 8,\;\;\; P(8) = 8$$
$$1\times2\times3\times4 = 24, \;\;\; ...
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votes
1answer
24 views
correlation of product with its normally distributed factors
If x and y are normally dist. with standard deviation of 10%, and they are independent, then their product X.Y is 71% correlated with Y (or X).
I can show this empirically, but how to I prove it in ...
3
votes
1answer
32 views
Probability $k$ bins are non-empty
The following problem arises in the analysis of Bloom filters.
Consider $m$ bins and $N=nk$ balls placed uniformly and independently at random into the bins. A query chooses $k$ bins uniformly and ...
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0answers
20 views
Generalizing the expression for the probability in of having exactly $k$ balls in a bin when $m$ balls are thrown in $n$ bins
The following lecture notes (due to Shuchi Chawla from U. Wisconsin) covers a set of randomized load balancing calculations for the scenario in which one is throwing $m$ balls into $n$ bins: ...
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0answers
26 views
Please show me that with which formula, I can calculate pooled variance for unequal population variance?
When equal population variances, I can calculate pooled variance (as like part-b)
But when unequal population variances, how to calculate pooled variance ( as like part-e)(also I underlined it ...
2
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1answer
35 views
Total probabilities of being admitted to any university
Let's provide an hypothetical situation in which a student applies to 10 different universities whose number of applicants, admissions and admission rate you can see in the table below.
...
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0answers
13 views
sampling schemes for binomial distribution
Two acceptance sampling schemes, A and B, are proposed for deciding whether or not to accept a large batch of items from a production process in which 5% of the items produced are defective. Scheme A: ...
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0answers
22 views
Is there an introduction to probability and statistics that balances frequentist and bayesian views?
Perhaps, roughly, I might be described as advanced undergraduate regarding mathematics. However, I have not learned statistics and have only learned elementary probability. Does there exist a book or ...
2
votes
2answers
176 views
Sum of variance
Say I make 3 independent experiments and these are the outputs
O/P of 1 st exp : 1,2,3
O/P of 2 nd exp : 4,5,6
O/P of 3 rd exp : 7,8,9
In general Var( A+B+C) = Var( A ) + Var( B ) + Var ( C )
In ...
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1answer
53 views
What makes the Var(x)+Var(y)=var(x+y) property important?
What makes the Var(x)+Var(y)=var(x+y) property important?
It was taught in my statistics class
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0answers
12 views
Question regarding Iteratively reweighted least squares?
If we have a set of data and then we want to find Iteratively reweighted least squares we know we have to use a weighting function. But I'm not sure how to find that weight corresponding to the data. ...
1
vote
1answer
24 views
Recovering random variable from its moments
The problem is: can you recover a distribution of random variable if you know all its moments?
My first guess was to use moment-generating function (MGF). It is known that if two random variables ...
3
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0answers
37 views
+50
Neyman-Pearson lemma on Normal distribution
We've got a random sample of iid $X_1,\dots,X_n$. We're testing the mean of $X \sim \mathcal{N}(\mu,\sigma^2)$, where $\sigma^2$ is known. The size of the test $\alpha=0.05$.
$H_0: \mu=0$
$H_1: ...
1
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0answers
24 views
Convergence in distribution and convergence of expectation.
Let $\{X_n\}_{n\geq1}$ and $\{Y_n\}_{n\geq1}$ be two sequences of uniformly integrable iid random variables with distributions $F_n(x)$ and $G_n(x)$, respectively. If
$$|F_n(x)-G_n(x)|\leq ...
1
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1answer
59 views
What will be the count of “4cardstraight” in a Poker Game
Let's modify the poker hand(fake): "4cardstraight" -- i.e. a straight, but with only 4 cards in a row instead of 5.
Rules of the game:
A hand is counted as a "4cardstraight" if it includes 4 ...
2
votes
2answers
56 views
Proving that $P\left(\bigcup_{j=1}^n E_j^{(n)}\right)\sim\sum_{j=1}^n P(E_j^{(n)})$ for independent events $E_j^{(n)}$
For arbitrary events $\{E_j, 1\le j\le n\}$, we have
$$P\left(\bigcup_{j=1}^n E_j\right)\ge\sum_{j=1}^n P(E_j) - \sum_{1\le j < k \le n} P(E_jE_k)$$
If $\forall n: \{E_j^{(n)}, 1\le ...
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1answer
26 views
Simple Expect Value Exercise
Question: We have $9$ coins, $1$ of them is false (lighter). We divide them up in pairs (with one left) and weigh them (that is taking two in a balance and seeing if one of them is lighter). What is ...
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1answer
32 views
A conditional probability problem on coupon collection
Suppose that there are $n$ types of coupons, and that the type of each new coupon obtained is independent of past selections and is equally likely to be any of $n$ types. Suppose one continues ...
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2answers
51 views
Random Poisson Sum of Random Variables with known distribution
I am trying to get a closed form expression for the expected value of the following summation of RVs: $\sum_{i=1}^{Y} X_{i}$, where $Y$ is Poisson distributed with parameter $\lambda$ and $ X_{i} $ ...
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0answers
14 views
The probability to be attacked with a particular strategy in a P2P network
In a P2P network of $n$ nodes, a fully connected graph, there are $3$ kinds of nodes, using strategy $A$, $B$, $C$. with probability $P_a$, $P_b$ and $P_c$.
The payoff matrix is the same of Rock Paper ...
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1answer
22 views
this is regarding exponentials distribution
In an office building, the lift breaks down randomly at a mean rate of 3 times per week. The random variable X represents the time in days between successive lift breakdowns.
(i) Calculate the ...
1
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0answers
33 views
Intuitive agument in case of a problem on Gambler's ruin
We have a gambler who at each step wins and loses $1$ dollar with probability $p$ and $1-p$ respectively. The game ends when he loses everything or wins $m$ dollar. Now starting with $i$ dollar the ...
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0answers
17 views
Comparison of Probabilities of Getting a Formula in Different Notational Schemes
I played the first two WFF 'N Proof games of the WFF 'N Proof kit tonight with a friend, and on my way home I started thinking. Suppose we have the set of variables, constants, and truth-functions ...
2
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1answer
31 views
A probability problem on coin toss
A collection of $n$ coins is flipped. The outcomes are independent, and the $i$-th coin comes up heads with probability $\alpha_i, i=1, \dots, n$. Suppose that for some value of $j, 1 \leq j \leq n, ...
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1answer
45 views
Expectation of a sum of independent and identical exponential variables
Suppose $c>0$. Let $X_i$ be independently and identically distributed exponential variables with parameter $\lambda >0$ and $$S_n=\sum_{i=1}^n X_i.$$ Let $I=\min \{i: X_i >c\}$. How do I show ...
1
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1answer
20 views
Computing the expected value of a matrix?
This question is about finding a covariance matrix and I wasn't sure about the final step.
Given a standard $d$-dimensional normal RVec $X=(X_1,\ldots,X_d)$ has i.i.d components $X_j\sim N(0,1)$. ...
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0answers
18 views
A version of coupon collector problem
Suppose each new coupon is, independent of the past, a type $i$ coupon with probability $p_i$. A total of $n$ coupons is to be collected. Let $A_i$ be event that there is at least one type $i$ in ...
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1answer
37 views
Probability on product spaces
I am having some trouble, more of an argument with someone else, about a simple question regarding product spaces.
Let $X_1,X_2,\dots,X_n$ a set of independent and identically distributed random ...
5
votes
4answers
77 views
Understanding Bayes' Theorem
I worked through some examples of Bayes' Theorem and now was reading the proof.
Bayes' Theorem states the following:
Suppose that the sample space S is partitioned into disjoint subsets $B_1, ...
3
votes
3answers
137 views
Probability of losing packets
I am currently enrolled in an Intro to Networking course and I have been studying for an upcoming exam by doing practice problems in the course textbook. I came across this question that stumped me. ...
1
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1answer
20 views
What is the purpose to define different moments on a distribution?
What is the purpose to define different moments on a distribution?
The first moment is the expectation value of a function, what about the other?
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1answer
17 views
easy question about conditional expectations
I got a quite easy exercise I just don't get.
Let P be a probability measure, $\frac{dQ}{dP}=Z$, $Z>0$ a.s. and $E[Z]=1$, hence Q is an equivalent proba-measure to P. Then I shall prove that for a ...
1
vote
1answer
312 views
Conditional Probability of Binomial Distribution
What is the probability of tossing $k_2$ heads in $n_2$ trials conditional that in the first $n_1$ attempts there were $k_2$ heads? Assume the probability of heads is $p$.
How does that change when I ...
5
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1answer
214 views
Expectation Problem (Khintchine's Inequality)
As a reference, this is Problem 6.2 in Albiac and Kalton's Topics in Banach Space Theory
The question involves a direct proof of Khintchine's Inequality. In part (1), we are to prove that ...
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0answers
18 views
The distribution of balls in bins as a function of the number of balls [duplicate]
If I toss $m$ balls into $N$ bins, as a function of $m$, what distribution of balls and bins should I expect as a function of $m$? What would be the mean largest number of balls in a particular bin?
