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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6 views

How are probabilities defined?

This stray thought has been bothering me for the past week. It seems that all probabilities and percentages are defined using the extremes 0% and 100%. Where: 0% is the probability that something ...
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0answers
4 views

Is $F_n\to F_{\infty}$ equivalent to $\lim_{n\to\infty}\int\phi dF_n=\int\phi dF_{\infty}$ for every $\phi \in C(R)$?

If $F_1,F_2,...,F_{\infty}$ are distribution functions. Is $F_n\to F_{\infty}$ equivalent to $\lim_{n\to\infty}\int\phi dF_n=\int\phi dF_{\infty}$ for every $\phi \in C(R)$? I intuitively think this ...
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2answers
31 views

Definition of $f \vee g$ and $f \wedge g$

In Olav Kallenberg's Foundations of Modern Probability he uses the notation $f \vee g$ and $f \wedge g$ where $f, g$ are two functions from a set $\Omega$ to $\mathbb{R}$. What does this notation ...
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0answers
12 views

Not sure what formula to use? (what to solve for?)

The question states, "The weight of people in a certain pacific island is normally distributed with a mean of 175 lb. and a standard deviation of 33 lb. They want to design a one-person canoe that ...
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0answers
4 views

Can we simplify the conditional covariance $\mathbb{V}[(X\:Y\:Z)|X+Y+Z=1]$?

Given random variables $X,Y,Z$ on a probability space, can I write the conditional covariance matrix $$\mathbb{V}\left[ \left(\begin{array}{c}X\\Y\\Z\end{array}\right) \Bigg|X+Y+Z=1\right]$$ as a ...
-1
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1answer
15 views

Variance formula in terms of the CDF for a continuous nonnegative random variable.

Is there a formula for the variance of a (continuous, non-negative) random variable in terms of its CDF? The only place I saw such formula was is Wikipedia's page for the Variance ...
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1answer
16 views

A coin Toss conditional probability question

I'm unable to solve this. Two players, A and B alternatively toss a fair coin (A tosses the coin first, then B tosses the coin, then A, then B and so on). The sequence of heads and tails is recorded. ...
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1answer
24 views

CDF of $X=\min\{X_1, X_1\cdot X_2\}$

I have random variable (RV) $X$ where $X=\min\{X_1, X_1\cdot X_2\}$. Further, $X_1$ and $X_2$ are independent but not identical RVs with exponential distributions, i.e., ...
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0answers
10 views

Number of edges in randomly induced graph

If I have a simple graph $G$ with $n$ vertices and $m$ edges, then I want to create a randomly induced graph $G_x$ by selecting vertices with a probability of $n/2m$. The edges of $G_x$ are defined to ...
0
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1answer
19 views

Expectation of the maximum of n random variables?

Let's say we have $n$ independent random variables, each variable equally likely to take any value in the interval $[0,1]$. What is the expectation of the maximum of these $n$ random variables? ...
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1answer
24 views

construction of Martingales

Consider a random sample of independent and identically distributed random variables with mean 1 . Consider another random variable which is the product of the first n of such random variables as ...
3
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1answer
39 views

What is the probability that I have seen every time on the clock?

Assuming a digital clock shows only hours and minutes, there are 1440 different times it may show. If you checked the time on 35000 independent occasions, what is the probability that you would have ...
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0answers
16 views

Sampling substrings of a beaded necklace to determine the necklace composition

I have a necklace composed of 100 beads, where each bead is one of 13 colors. If I am only able to look at one 4 bead sub-sequence at a time (connected, as they would be on the necklace) , how many ...
1
vote
1answer
30 views

Writing the expected value of a random variable in terms of its cumulative distribution function

My professor said that an alternative expression for the expected value of a random variable can be written as: $$ E[X] = \int_{0}^{\infty} (1-F_X(x)) \, dx - \int_{-\infty}^0 F_X(x) \, dx $$ No ...
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2answers
56 views

Is this an application of the Birthday problem?

Let's say there is some positive integer n that is somewhere between 0 and N (also a positive integer). I tell the program to start generating random (or pseudo-random) number pairs (modulo N) and ...
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0answers
17 views

how to find joint probability distribution under conditioning [on hold]

given two pdf of random variable X,Y. Whether it is possible to find joint pdf of x and y under the condition x>y. Is it possible to consider x and y are independent.
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0answers
27 views

Show that If E ⊂ F, then P(E) ≤ P(F).

I am trying to solve this following problem If E ⊂ F, then P(E) ≤ P(F). But i am having no idea where to start from. Can anyone please help me with that? Thanks
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2answers
39 views

Suppose $\xi_1, \xi_2,\ldots$ are i.i.d. random variables with mean $\mu$, variance $\sigma^2$. Form the random sum $S_{N} = \xi_{1}+\cdots+\xi_{N}$.

(a) Derive the mean and variance of $S_{N}$ when $N$ has Poisson distribution with parameter $\lambda$. So far, for the mean, I have the following: $E[S_{N}] = E[E[S_{N}\mid N=n]]$ $$ = ...
1
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0answers
31 views

Polynomial Interpolation When part of $y_i$'s are Shuffled

Hypothesis: Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and ...
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1answer
24 views

Probability - conditional [on hold]

The probability that bulbs are detected faulty if they are defective is 0.95 and the probability that bulbs are declared fine if in fact they are fine is 0.97. If 0.05 of the bulbs are faulty, what is ...
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1answer
22 views

How to prove that the set of all exchangeable events is a sigma-algebra?

Let $ {X_n}_n $ be sequence of identical R.Vs Mark by S the set of all sequences available from it. An exchangeable event is $E\subset S $ which is not sensitive for finite permutations. ...
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0answers
19 views

The probability that two matrix vector products are equal

Consider a random $n$ by $n$ circulant matrix $M$ whose first row entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first ...
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1answer
28 views

Probability -dividing into groups

In how many ways can 12 people be separated into 3 groups of 4 if the 12 comprises 6 pairs of partners? We must keep partners in the same group, but we do not distinguish between the group $(a, b, c, ...
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1answer
22 views

In how many ways that the letters of ENTERTAINMENT are arranged in a row where two Es are together and one is apart [on hold]

In how many ways that the letters of ENTERTAINMENT are arranged in a row where two Es are together and one is apart??
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2answers
48 views

dice probability - same 2 dice in 6 dice rolls

I have this simple probability problem that I am not sure I solved correctly. I am not interested in formulas, but rather the thought process of how to solve it. Suppose we roll six 6-sided dice that ...
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3answers
41 views

Random variable with 2 distribution functions

Just a question here, Given a random variable $X$ defined in a probability space, is it possible to have more than one distribution function $F$ ?
1
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1answer
33 views

Probability that a year contains 53 Mondays

The question: Find a probability that a year chosen at random has 53 Mondays. Now I know in a non-leap year, probability of getting 53 Mondays is $\frac{1}{7}$ and in a leap year, probability of ...
0
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1answer
33 views

Drunk Passenger Probability question [duplicate]

I don't know how to solve this question. A line of 100 airline passengers are waiting to board a plane. They each hold a ticket to one of the 100 seats on the flight. For convenience, lets say that ...
0
votes
1answer
16 views

Derivative of Poisson that approximates Binomial

Instead of a standard urn ball problem, I have many urns and balls. Many. One might say, a continuum of balls $B$ and urns $U$. The likelihood of a single urn having $x$ matches is, under the ...
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3answers
29 views

The dice is rolled 10 times and the results are added with given conditions.

Q: A dice has one of the first 6 prime number on each its six sides ,with no two sides having the same number .the dice is rolled 10 times and the results added.the addition is most likely to be ...
2
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1answer
24 views

Law of a random variable (characterization)

If $X$ is a real random variable defined on $(\Omega,\mathcal{F},\mathbf{P})$ then there exist several characterizations of the law of $X$ being $\mu$ : $X \sim \mu$ if and only if for every ...
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4answers
696 views

Probability that a natural number is a sum of two squares?

Some natural numbers can be expressed as a sum of two squares: $$2=1^2+1^2$$ $$25=3^2+4^2$$ $$50=7^2+1^2$$ If one chooses a random natural number, what would be the probability that that number is a ...
-2
votes
1answer
30 views

Expected number of customers sitting on correct places [on hold]

In a shop customers are given a seat number before entering the shop in the order 1,2,3,...,n but after entering the shop they sit in a random order not related to their seat number. what is the ...
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2answers
16 views

how Find the probability that the committee will consist of the following all dentists

A committee of four people is to be formed from six doctors and eight dentists. Find the probability that the committee will consist of only dentists
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1answer
28 views

Determining bounds for change sum of continuous r.v.'s

I'm trying to understand how to determine the bounds when computing the sum of continuous random variables. Here is a sample question: X and Y have the following joint pdf: $f_{X,Y}(x,y) = 4xy, 0 ...
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votes
1answer
54 views

Probability of randomly selecting one student from each of three cities [on hold]

The geographical distribution of hometown of some 80 students at DLSU-D is given as: 50 from Cavite, 10 from Laguna, and 20 from Manila. Suppose three students are selected. Find the probability that ...
1
vote
1answer
23 views

Expected value of function of minimum between two random variables

Two independent random variable $X,Y$ are distributed on $[0,\infty)$ according to the cumulative distribution function $F(x)=1-(x+1)^{-2}$. Let $Z=\min(X,Y)$. Determine $E\left[\frac{Z}{Z+2}\right].$ ...
1
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0answers
20 views

An inequality related to supermartingale?

Let $X_n\ge0,n\ge0$, be a supermartingale. Show that $CP(\sup X_n>C)\le EX_0$. I tried to use the inequality supermartingale satisfies, which is $E(X_n|\cal {F_{n-1}})$$\le X_{n-1}$. However, ...
0
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0answers
29 views

How can I formed as below permutation problem

Hi I am writing a program and i encouraged the below permutation problem and need your help. There are 4 boxes: 3 of them have 2 balls The one box has 1 balls. The question is what is the ...
0
votes
2answers
26 views

Anagrams contained within random strings

What is the probability that a random string of length $n$ will contain an anagram of a shorter string of length $k$? E.g., you generate a string of 50 random letters, repetitions allowed, what are ...
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0answers
28 views

Probability distributions associated with Markov chain

Let's say I have a Markov chain, with all the transition probabilities known, and there's a cost associated with each transition. The cost for transitioning from node $a$ to node $b$ is given by the ...
0
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1answer
20 views

5-Card Poker Two-Pair Probability Calculation

Question: What is the probability that 5 cards dealt from a deck of 52 (without replacement) contain exactly two distinct pairs (meaning no full house)? Solution: ...
0
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1answer
27 views

How to calculate the odds of a 5x5 Bingo game?

I don't have a mathematics background, but am trying to calculate what the theoretical odds of winning a 5x5 bingo game is if 5 numbers are drawn. Eg board: ...
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1answer
25 views

Prove that X has a chi square distribution

If $X_1,\dots ,X_{30}\sim N(1,\sigma^2)$ and $\hat \sigma^2 = \frac{\sum(X_i-1)^2}{30}, $ then show that $30\,\hat σ^2 /σ^2$ has a chi-square distribution with $30$ degrees of freedom.
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0answers
21 views

Difference between the “Hazard Rate” and the “Killing Function” of a diffusion model?

I posted this question on Cross Validated - but I think it applies here too. Also, it increases the chances of the question being answered. Link here If this is not acceptable - administrators ...
2
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0answers
17 views

a conceptual question on markov chain [duplicate]

Suppose $\{X_n,n\ge 0\}$ and $\{Y_n,n\ge0\}$ are two independent discrete-time markov chains (DTMC) with state space $S=\{0,1,2,\ldots\}$. Prove or give a counterexample to: $\{X_n+Y_n,n\ge 0\}$ is ...
0
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1answer
26 views

Interpretation of the negative binomial and geometric distributions

I am having trouble putting together the way these distributions work. It doesn't matter whether we speak of the support space in terms of number of trials or failures. Specifically what variable is ...
2
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1answer
22 views

An Example of sequence of R.V with $E(X_n) = X_0$ but $E(X_n^{1/2}) \to 0$

I need an example of $\{X_n\}_n$ be a sequence of nonnegative, random variables, with the same finite expected value $E(X_n)=\mu_0$, that obeys: $E(\sqrt{X_n})>E(\sqrt{X_{n+1}})>\dots \to 0$
2
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1answer
16 views

Expectation of max absolute value of a Gaussian vector

Let $X$ be a joint Gaussian vector of dimension $k$ with zero mean and covariance matrix $K$ (where $K$ may not be diagonal). I am interested in sharp estimates on $$\mathbb{E}\max_{i=1,2,\ldots,k} ...
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2answers
30 views

What is the probability that both the numbers are odd with given conditions?

2 Numbers are selected at random from the integers 1 through 9.If the sum is even,find the probability that both the numbers are odd. My approach: A:Event of getting sum as even, B:Event of ...