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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17 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 ...
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1answer
17 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 obeyes: $E(\sqrt{X_n})>E(\sqrt{X_{n+1}})>\dots \to 0$
2
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1answer
12 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
28 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 ...
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0answers
49 views

Monty Hall problem again (from Grimmet and Stirzaker)

Grimmet and Stirzaker Exercise 1.4.5.2 In a game show you have to choose one of three doors. One conceals a car, 2 conceal goats. You choose a door but the door is not opened immediately. Instead ...
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2answers
25 views

If $x_i$ is from a random sample is $Var(\bar x \mid x_i)=0$?

If $x_i$ is from a random sample, is the conditional variance of the mean (or the sum of squares, really any statistic based on $x$) just treated as a constant? I saw this in a OLS variance of a ...
2
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2answers
31 views

Infinite sequence of exponentially distributed random variables

Consider an infinite sequence of exponentially distributed random variables, $X_k$, where$ k \in \{1, \ldots, n\}$ with $\lambda = 1$. I am trying to evaluate: $$\lim_{n\to\infty} \frac{\max_{1 \leq ...
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1answer
32 views

Probability mass function (pmf)

Let's say we have an computer program with a loop. Inside the loop the program encounters a problem with probability p, in which case it increases a counter by 1 and prints out a warning. If the ...
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2answers
22 views

I need help with finding a probability [on hold]

Here is the question A bag contains 25 coins. There are 6 quarters,4 dimes,5 nickels,and the rest are pennies One coin is drawn at random. Find the probability that it will be a nickel
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0answers
22 views

Restriction over pdf such that an integral inequality holds $\int_{-\infty}^{+\infty}\left(F(x)-\frac{2}{3}\right)xf(x)dx\geq 0$

Let $f(x)$ be a pdf in $(-\infty,+\infty)$ and $F(x)$ it's cdf. Assume both are smooth. I need to find restrictions over the pdf such that the following inequality holds: ...
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0answers
20 views

Question related to Square Integrable Martingale

Let $X_n$ and $Y_n$ be martingales with $EX_n^2<\infty$ and $EY_n^2<\infty$ for all $n$. Show that $EX_nY_n-EX_0Y_0=\sum_{m=1}^nE(X_m-X_{m-1})(Y_m-Y_{m-1})$. I tried to expand the right ...
2
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2answers
27 views

Probability that two sets do not intersect

I'm trying to understand this simpler problem so I can apply the process to a more difficult homework problem. Let $U$ be a set with $n$ elements. Select $2$ independent random subsets $A_1, A_2 ...
3
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1answer
37 views

Law of Large Numbers - utility/difficulty of various versions.

This may or may not be an answer to Is there an easy proof that the set of $x \in [0,1]$ whose limit of proportion of 1's in binary expansion of $x$ does not exist has measure zero?, depending on ...
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0answers
24 views

Lower bound for $\Pi(n)$ - viability of probabilistic theory

Can somebody check the validity of my arguments below, and tell me why its wrong or right? Consider the sequence of non-negative integers. Let $a_0=0, a_1=1, ..., a_i=i,...$ Divisiblilty of $a_i$ ...
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1answer
35 views

equation for probability stumper?

Jane says she was in town for two consecutive days last week (7 days), but won't say more. For each given day, what was the probability she was in town that day? I know there are 6 possible ...
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2answers
32 views

I need to find the mean of students who got below the 70th percentile [on hold]

$50$ children took a test. $5$ children got $100\%$, another $10$ got $90\%$, and yet the average class score was $45\%$. What is the mean score for the children below the 70th percentile? a. ...
2
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1answer
22 views

Finding the distribution of the sum of n independent random variables having exponential distributions

My graduate level probability class asks us to calculate the distribution of the sum of n independent exponentially distributed random variables. I am trying to perform many convolutions but it gets ...
1
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1answer
30 views

How many ways can an integer $i$ appearing in a sequence with multiplicty at least $j$, be minimal

Let us construct an integer sequence of length $n$, where the integers are chosen from $\{1, 2, ..., k\}$, with i.i.d. uniform probability $\frac{1}{k}$. I want to compute the probability ($p_{ij}$) ...
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2answers
41 views

Help needed to solve probability problem

I am trying to solve the following problem. A fisherman is equally likely to go fishing at one of the three ponds A,B,C. The probability to catch fish if he cast his rod at pond A is 0.4 , at pond B ...
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0answers
23 views

In an incomplete market does every payoff function admit at least two arbitrage-free pricings?

Consider an arbitrage-free (not necessarily complete) market. Prove or disprove the following assertion. If the market is incomplete, then every payoff function $A : \Omega \rightarrow \mathbb{R}$ ...
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1answer
59 views

How to remember these probability results?

If $A,B$ and $C$ are $3$ events, then $P$(Exactly one of $A,B,C$ occurs)$=P(A)+P(B)+P(C)-2[P(A \cap B)+P(B \cap C)+P(A \cap C)]+3P(A \cap B \cap C)$ $P$(Exactly two of $A,B,C$ occur)$=P(A \cap ...
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1answer
22 views

Basic conditional probability question [on hold]

$\sum_{c}p(a|c)p(c|b)=p(a|b)$. Does this equation hold true? If it is true, how to prove it mathematically?
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4answers
59 views

What is the probability of drawing 3 balls such that none of them is red?

Given a bag containing $8\ \color{red}{red}$ balls and $4\ \color{green}{green}$ balls, what is the probability of drawing $3$ balls at random such that $\mathbf {none}$ of them are ...
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1answer
36 views

Expected % of heads flipping coins of different odds

So this is an analogy for a real world example but for simplicity. So if I were to flip a normal coin ten times I would expect heads 50% of the time or 5 head results. I could then compare this to the ...
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4answers
1k views

Minesweeper probability

I ran into the situation pictured in the minesweeper game below. Note that the picture is only a small section of the entire board. Note: The bottom right 1 is the bottom right corner tile of the ...
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votes
2answers
59 views

Probability that distance of two random points within a sphere is less than a constant

Two points are chosen at random within a sphere of radius $r$. How to calculate the probability that the distance of these two points is $< d$? My first approach was to divide the volume of a ...
-1
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0answers
10 views

Sampling of a changing mixture model

Let f, g, and h be probability density functions, and X, Y and Z be random variables respectively following f, g and h. The mixture model: ${\text h = \frac{f + g}{2}}$ states that the distribution ...
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3answers
58 views

Doubt about a probability excercise

I'm a statistics teacher at a college. One day a student came with a doubt about an exercise about probability. The text goes like this: A person has two boxes $A$ and $B$. In the first one has ...
-1
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1answer
16 views

Random Permutation Polynomial With Fixed Inputs

Assume we pick uniformly random a permutation polynomial, $T$, of degree one. we define all polynomials over $\mathbb{Z}_P$. We have fixed inputs $x_i$ (e.g. $x_i \in [1,100]$) My Question: Is ...
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0answers
25 views

Calculating Combinations / Permutations [on hold]

How do I calculate the number of outcomes as a whole of a series of individual tests with there own outcomes? For example, the best description I could think of would be: There are 10 tests and each ...
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1answer
21 views

Double integral proving that a function is a probability density

If $$g(x,y)=f(x+y)/(x+y)$$ for $x,y>0$ and $$\int_0^{\infty} f(z) \, dz = 1$$ How do you show that $$\int_0^{\infty} \int_0^{\infty} \frac{f(x+y)}{x+y} dx \, dy = 1$$ as well?
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0answers
18 views

Uniqueness of the transformation turning random variables into IID uniform

We have two random variable $X:\Omega \to \mathbb R $ and $Y: \Omega \to \mathbb R^d, d \in \mathbb N$, $F_Y$ is the density function of $Y$ and $F_{X|Y=y}$ is a regular density function of $X$ ...
1
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1answer
41 views

Properties of independence and conditional independence

Recently, I see some properties from conditional independence wiki page https://en.wikipedia.org/wiki/Conditional_independence I don't quite understand the properties of "Rules of conditional ...
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0answers
22 views

Convolution of independent but 'different' probability distributions

I have the following two probability distributions they relate to a particular ice-cream: ...
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0answers
17 views

Compute distribution in Hidden Markov models

Let $Z_1, Z_2, ..., Z_n$ be the latent variables, and $X_1, X_2, ... X_n$ be the observed ones in a hidden markov models. Let's assume that the parameters of the hidden Markov models are known: the ...
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3answers
63 views

Can someone give me real world example of uniform distribution [0,1] of a continuous random variable.

Can someone give me real world example of uniform distribution [0,1] of a continuous random variable, because I could not make out one.
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0answers
19 views

what is the intuition behind the SRSWOR formula?

I earlier asked about Slovin's Formula, and learned shortly thereafter that it was derived from this formula. $n=\dfrac{n_0}{1+\dfrac{n_0}{N}}$, Where $n_0=\dfrac{z^2p(1-p)}{e^2}$. So, breaking it ...
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0answers
20 views

Distributions of identical and distinct objects [on hold]

I'm having an issue figuring this problem out. I'm not sure how I should go about it exactly, all I know is that it needs to divided into stages, each its with its own set of cases. So here's is ...
1
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3answers
73 views

How to calculate the limit of $(\frac{x}{x+1})^x$

I am looking at the probability of losing $x$ games in a row, in a game where the probability of winning is $1/x$. (For example, if this is a fair casino game, what is the probability of losing $x$ ...
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0answers
21 views

Probability of me being one of a group

I've heard that in a certain country with a population of about 140,000,000, 200 people become missing on a daily basis. If I want to calculate the random probability that one of those people would ...
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0answers
17 views

what is formula to this eqution [(256)16]1/32+[(169)6]1/12 [on hold]

how to solve this equation [(256)16]1/32+[(169)6]1/12 what is formula of this? What is the closed form expression for this? What is the right domain for this Hamiltonian 2? what is the right ...
0
votes
1answer
21 views

Conditional Probability Equivalence

I am reading this paper here: http://www.utm.utoronto.ca/~weisber3/articles/SobervBJPS3.SP.pdf which claims on page 10 that $p(E \wedge [H_x \wedge \sim R]) + p(E \wedge [H_x \wedge E])= p(H_x \wedge ...
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2answers
27 views

Convergence Events with States

Ignatz repeatedly rolls a fair $6$-sided die. What is the probability that he rolls his first $5$ before he rolls his second (not necessarily distinct) even number? I don't know what to do about the ...
1
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1answer
25 views

3 Events, Conditional Independence

Given $A,B,C$ such that: $$ P(A\mid B),P(A\mid B^c),P(B\mid C),P(B\mid C^c) \text{ are known } $$ and that $A,C$ are conditionally independent given $B$, so that: $$ P(A\mid B\cap C)=P(A\mid ...
2
votes
3answers
61 views

How do mathematician make sense of “outcome” and “events” in probability?

One of the biggest challenge for me to understand probability is to make sense of this concept of outcomes and events. To put it plainly, it just doesn't feel like mathematics anymore when we talk ...
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3answers
50 views

Intuitive meaning of the probability density function at a point

I understand how to integrate probability density functions to find probability within a certain range. However, what I don't understand is what it would mean to set the variable (say x or y) to a ...
-2
votes
1answer
35 views

What time should Celia aim for in her sixth race to make the team? [on hold]

To be on the 1-km race team, Celia must have a mean time less than 5 min 50 sec in her 6 tryout races. Her times in 5 races are: 6 min 2 sec, 5 min 53 sec, 5 min 45 sec, 6 min, and 5 min 34 sec. What ...
1
vote
1answer
22 views

When randomly distributing n points amongst m people, what are the odds that one certain person will get a certain amount of points?

I'm mostly curious about how to find this in general, but the actual problem is with 20 points and 5 people. I know probability problems are very counterintuitive, and thus I was unsure after ...
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1answer
26 views

Probability with $n$ successes before $m$ failures

Independent trials resulting in a success with probability $p$ and a failure with probability $1 − p$ are performed. What is the probability that $n$ successes occur before $m$ failures? Given ...
0
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2answers
22 views

Mathematical Backing for Observations seen in Adding Independent Random Variables Together

So I have a function Y = F(N) that takes as an argument an integer number N and returns a summation of N sine-waves of different random parameters. I have plotted the results of two function calls ...