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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13 views

how to simplify this conditional expectation

Suppose $v_1,v_2,v_3$ are three random variables drawn independently from the same distribution $\mathrm{uniform}(0,1)$, is it correct that $E_3[v_3\mid v_1 < \max\{v_2,v_3\}, ...
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1answer
22 views

apply the law of total expectation

I'm a little bit confused about applying the law of total expectation. Suppose $v_1,v_2,v_3$ are three random variables drawn independently from the same distribution $\mathrm{uniform}(0,1)$, which ...
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0answers
18 views

Counterexample for $(Y_1 \perp Y_2) \mid (X_1, X_2) \Rightarrow (Y_1 \perp Y_2) | X_1$?

Let $(Y_1 \perp Y_2) \mid (X_1, X_2)$ mean that random variables $Y_1$ and $Y_2$ are conditionally independent on $(X_1, X_2)$. Either is there a counterexample for $(Y_1 \perp Y_2) \mid (X_1, X_2) ...
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0answers
16 views

Confidence interval for differenciation

I will also ask the question here, because I realized it can also be a logical/probability problem. I'm a chemist so my question will be experience-oriented. Lets' say I have 5 molecules (A, B, C, D, ...
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0answers
15 views

Probability Density following affine transformation

Suppose $X$ is a random variable in $R^n$ and $Y=a^{T}X+b ∊ R$. If $f_X$ is the density of $X$, then what (and how!) can I obtain $f_Y$ the density of $Y$? It is assumed that $a\neq 0$. I saw the ...
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0answers
18 views

What is the Cumulative Distribution Function of $a/x^b$?

I was just wondering what the CDF of $$\frac{a}{x^b}$$ would be? $a$ and $b$ are positive constants and $x \ge 0$. It's puzzling me even though it seems trivial. Thanks for your help.
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2answers
21 views

Probability of single digits from coin tosses

Let's say that I wanted to generate 4 random numbers using a coin toss. I could toss the (unbiased) coin 4 times to generate one of 16 possible numbers (e.g. TTHH=0011=3) and just ignore any results ...
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0answers
19 views

Finding n for a given P of a Bernoulli trial

I'm randomly sampling $N$ items and I want to find $n$ such that I have a probability $P$ that I'll miss one. Practically, I'd select $P$ to be something like $10^{-12}$ so I'm almost assured to ...
2
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0answers
35 views

Probability of an Indisputable winner at Texas Holdem

What are the fraction of hands that can be classified as "indisputable winners" after the river is revealed in Texas Holdem? An indisputable winner is a hand that cannot lose. A clear example would ...
2
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3answers
45 views

Adding two discrete distributions

I am taking a probability course and I am having trouble adding two discrete distributions. The two distributions given are: $X$ has a discrete uniform distribution on the integers $0,1, ... ,9$. ...
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1answer
29 views

Russian Roulette consecutive bullets

You are playing a game of Russian Roulette. If instead of one bullet, two bullets are randomly put in the chamber. Your opponent played the first and he was alive after the first trigger pull. You ...
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0answers
26 views

Conditional Probability Error Question

Let $P_0 = 0.25,$ $P_1 = 0.35$, $P_2 = 0.25,$ and $P_3 = 0.15.$ What is the probability of more than one error? I thought to sum $P_1, P_2, P_3$ together but that doesn't seem to work. What formula ...
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1answer
13 views

Query about non-singular transformation of vectors

Suppose we are given a probability function, P (x^T (Y-z)≥0) , where ‘x’ is a vector, ‘Y’ is a random variable and ‘z’ is a known value. Now, suppose, we make a non-singular transformation w=Ax, ...
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0answers
11 views

a problem involving binary entropy function

let $\alpha<1/2$ such that $2^{H(\alpha)}\le 2^{1-\epsilon}$,when $H$ is binary entropy function. how can i prove that then we have: $2^{n(1-\epsilon)}\ge \sum\limits_{i\le \alpha n } {n \choose ...
2
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2answers
26 views

Count ways to place $n$ identical balls into $n$ urns so that exactly one urn is empty?

How many ways are there to plane n indistinguishable balls into n urns so that exactly one urn is empty? Why is the answer for this question n(n-1)?
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1answer
28 views

Probability of a combination of two distinct numbers chosen from $1, \dots, 28$ [on hold]

If two distinct numbers are taken from $1,2,3, \dots, 28$, what is the probability that their sum is less than $13$?
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1answer
44 views

Can somebody integrate this function for me? [on hold]

This is the function. $\frac{1}{6.08 \cdot \sqrt{2\pi}}\exp\left(-\frac{(x-10.75)^2}{2 \cdot 6.08^2}\right)$ Thanks in advance!
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2answers
27 views

In a game of poker, what is the probability that a five-card hand will contain…

In a game of poker, what is the probability that a five-card hand will contain (a) a straight (5 cards in unbroken numerical sequence) and (b) four of a kind. Solution given for (a) is $10(4^5 - 4)/ ...
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1answer
12 views

Finding out the percentage points.( F - Distribution).

How to find the values of these $x_1$ and $x_2$ , given , $P(x_1<F_{7,7}<x_2) = 0.90$ , using the F-Distribution tables.. Can anyone provide me a hint for this ?
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3answers
24 views

Chance of getting tails in the second throw with abnormal coin.

You have one coin that is normal, and one coin that have "tails" on both sides. Then you choose a coin without knowing which one, and you flip it. You get tails and decide to flip it again, what are ...
2
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1answer
43 views

Why is the measure of a boundary of an open ball positive in only a countable number of cases?

Let $X$ be a Polish (complete separable metric) space and $\mathbb{P}$ a Borel probability measure on $X$. Let $x_1, x_2, \ldots$ be a sequence of points dense in $X$. How can you prove that there is ...
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0answers
12 views

Calculating the Shannon information of drawing equal no. of cards

One card is drawn each from a $k$ deck of 52 cards where $k$ is a multiple of $52$. I need to prove that information of an outcome where each card appears the same number of times tends to ...
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1answer
21 views

Probability - branching

A population starts with 1 member: at t=1 , it can either divide with probability of p or dies with probability of 1-p. If it divides, then both of its children behave independently with the same ...
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1answer
22 views

Classical Probability and Combinatorics

Shuffle a standard deck of cards and cut it into three piles. What is the probability that a face card will turn up on top of one of the piles? There are 12 face cards (four jacks, four queens and ...
2
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1answer
17 views

Statistical bias and the probability of an outcome.

A town referendum has occurred. The question posed to voters was YES or NO on a local law. There were 3 methods of voting: Electronic machine (voting booths), absentee ballot, and affidavit ballot. ...
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2answers
23 views

Probability-Bayes theorem

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 0.5% of the bulbs are faulty, what is ...
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2answers
35 views

Russian Roulette probability question

You are playing a game of Russian Roulette. If instead of one bullet, two bullets are randomly put in the chamber. Your opponent played the first and he was alive after the first trigger pull. You ...
0
votes
1answer
31 views

to verify a relation involving conditional probability with an example

we have a relation P(A/complement(B)) = (P(A) - P(A/B)P(B))/(1-P(B)). This equation satisfies for A and B except P(B)=1. If X,Z are independent exponential RV with parameter $\lambda$1 = 1/10.39 ...
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0answers
15 views

Why distribution of multiple recursive random number generators is uniform?

I was reading the article of L'Ecuyer on random number generation. The title of this article is "Uniform Random Number Generation". One of the proposed PRNGs there, is multiple recursive random ...
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0answers
23 views

How to calculate multiple dice turns?

Assume we have a single dice , with 10 faces. We want the dice to show up "[1] face" 10 times And those 10 times must be arrange one after another. Like this. Startgame .....10 , 2 , 4 , 5 , 1 ...
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1answer
12 views

Bounding probability based on binary values

I've been reading this paper on probabilistic logic: http://ai.stanford.edu/~nilsson/OnlinePubs-Nils/PublishedPapers/problogic.pdf On page 76 theres a 3d diagram and Nilsson mentions the bounds on ...
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1answer
27 views

Calculating conditional probability of discrete uniform r.v.

X is a discrete uniform random variable on $\{a, a+1, a+2, ... , b\}$ with mean 7 and variance 4. Find $Pr[X \leq 6| X > 4]$ I'm not familiar with the discrete uniform distribution. I was ...
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3answers
37 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
20 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 ...
2
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0answers
27 views

Help provide some hint on a stirling approximation problem.

Use Stirling approximation to show $\mathop {\lim }\limits_{x \nearrow \infty } {e^{ - x}}\sum\limits_{x + a\sqrt x \le n \le x + b\sqrt x } {\frac{{{x^n}}}{{n!}}} = \int_a^b {\frac{{{e^{ - ...
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2answers
39 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
19 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
25 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 we 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 ...
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votes
1answer
16 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 ...
0
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1answer
29 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. ...
1
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1answer
30 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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1answer
18 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 ...
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1answer
22 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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2answers
40 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
41 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
23 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 ...
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1answer
33 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
59 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
18 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
28 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