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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24 views

Expectation maximization modeling

The setting of my problem is as follows: we have a set of questions $T_1,..T_n$ and a set of workers $W_1,W_2,...,W_m$ and a matrix where a cell $c[i,j]$ is the answer of question $T_i$ by worker ...
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0answers
120 views

Probability Theory: Cramer Transform Issue and deviation probabilities theory.

Let $\xi$ be random variable with probability distribution $F_\xi$. Let $\tilde\xi$ be random variable with probability distribution $\tilde F_\xi$ $I_{\xi}^t(x) = \int \limits_{-\infty}^x ...
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0answers
93 views

Poisson Processes and “Cooldown” Effects

This question is a spin on the traditional Poisson Process. Let's assume that I have a machine that has the ability to turn on a light. It does so by following a Poisson distribution with some ...
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1answer
145 views

Probability Mass Function of largest number selected without replacement

Objects are numbered 1 to $x$. If $y$ of them are randomly selected without replacement, find the pmf of $R_{yx}$, where $R_{yx}$ is the largest number selected and $y<x$. So far I have: ...
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6answers
645 views

Roulette tactics probability

In the American Roulette wheel, the winning odds of betting colours(black/red) is 47.36%. Consider this method, if the minimum single bet is \$10 and if I were to bet \$10 and double my next bet of ...
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2answers
61 views

Determine $P(X=k)$ if given a generating function

Simple question: If I have a non-negative, integer valued, random variable $X$ that has a generating function $g_x(t)=log(\frac{1}{1-qt})$ how would I go about determining $P(X=k)$ for {k = 0, 1, 2, ...
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1answer
67 views

Probability of getting loops

You are given $3$ bits of lace, if ends are tied together at random, what is the probability that you end up with $2$ loops? Generalise this for $n$ bits of lace. Ok so clearly I have 6 ends to play ...
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1answer
48 views

Probability and joint distribution

Suppose $x$ and $y$ are jointly normal with known parameters. I need to find the probability of the event in the parenthesis: $\Pr\left(x>f(y; a),y>g(x;b)\right)$. The function $f(\cdot)$ is ...
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1answer
54 views

Finding range, partiton and finding the dependency of a random variable

How to do this question part (a) range(X) =(2,3,4,5,....12) right the range is usually from 2 to 12 range(y) =(2,3,4,5,6) the range is usually from two to 6 right? range(z) = (0,1,2,3,4,5) ...
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2answers
258 views

Birthday paradox: Comparing the original version with the same-birthday-as-you version

The birthday paradox itself is well known. I am only interested in a small aspect here: The number of pairings in the original problem is $${23 \choose 2} = \frac{23 \cdot 22}{2}=253$$ Another ...
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2answers
68 views

Computation of mean and variance of $T$ with $P(T>t)$ given

Let's say we have: $$ P(T>t) = ae^{-L t} + (1-a) e^{-ut}, $$ where $T$ is the duration, $t \geq 0$; and $0 \leq a \leq 1$, $L > 0$, $u > 0$ are constants. What's here, in this case, the ...
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1answer
1k views

Probability: Determining Which Phone Plan Is Better

A consumer is trying to decide between two long-distance calling plans. The first one charges a flat rate of $10$ cents per minute, whereas the second charges a flat rate of $99$ cents for calls up ...
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1answer
195 views

Expectation value of $1/x$

Given a random variable $x$ which is assumed to follow a Gaussian distribution $x \sim N( \mu, \sigma^2 )$ and $x$ is further known to be positive, I am interested in the following expectation value: ...
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2answers
113 views

Probability question, not sure if I'm doing this right… at least vs exactly.

So I'm trying to figure a few things out with probability/counting. These a probability questions, but my understanding of the counting behind them is a little fuzzy still. For example, The ...
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0answers
248 views

Derivation of softmax function

I'm reading Bishop's book on Pattern Recognition and machine learning and I wanted to reproduce a calculation for the softmax function, also known as normalized exponential. Basically, the calculation ...
2
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1answer
21 views

Equality Question & Probability

Y is a binomial distribution(100, 0.5) P(|Y-50|>= 2) = 1 - P(49<= Y <= 51) ... i was reading these solutions in an exercise and was wondering how they came up with the next line. I thought the ...
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2answers
94 views

Probability to win between 7 and 10 games of the next 15 games

The probability that the Mets win is .8. What is the probability they win between 7 and 10 games of the next 15 games? Please help. Thank you.
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0answers
308 views

Unknown variance hypothesis testing for normal random variables

Let $X_1,...,X_n$ be iid random variables each with a $N(\mu_0,\sigma^2)$ distribution, where $\mu_0$ is known and $\sigma^2$ is unknown. Find the best (most powerful) test of size at most $\alpha$ ...
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2answers
54 views

r.v. Law of the min

On a probability space $(\Omega, A, P)$, and given a r.v. $(X,Y)$ with values in $R^2$. If the law of $(X,Y)$ is $\lambda \mu e^{-\lambda x - \mu y } 1_{R^2_+} (x,y) dx dy$, what is the law of the min ...
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2answers
140 views

Probability of finding adjacent colored squares in a line of white squares

So this question has a small science background, but the problem itself is purely mathematical. Consider a one-dimensional row of squares, some are white, some are blue. The blue squares represent ...
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0answers
82 views

Conditional distributions of (higher-order) autoregressive Markov processes

If we specify an $p$-th order autoregressive process in discrete time by its transition distribution $F_{t|t-1,\ldots,t-p}$, what can be said about lower order conditional distribution where we ...
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1answer
73 views

What does multilinear function mean?

A draft research paper claims that $Q(p)=1-p_1 p_2 p_3 p_4 - p_2 p_3 p_6 p_7-p_1p_2$ is multilinear where $p_i = \mathbb P(e_i)$ and $e_i$ is a basic event of a component to fail. I have learnt in LP ...
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1answer
976 views

Exponential Distribution Of Rainfall

Data collected at Toronto Pearson International Airport suggests that an exponential distribution with mean value 2.725 hours is a good model for rainfall duration (Urban Stormwater Management ...
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2answers
337 views

Almost sure convergence of random variable

I see a lot of examples of limit theorems in terms of functions, and sequences of functions. But I think the transition from the general measure space to the probability space ...
5
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2answers
138 views

Probability of coins flipping

Suppose a coin has probability $p$ for heads and $(1-p)$ for tail. Let $P_{k,p}$ be the probability that in $N$ flips there is a sequence of consecutive heads of length greater than or equal to $k$. ...
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1answer
80 views

Poisson Distribution, not sure how to include several events

This is the very last question on an assignment sheet and for some reason I can't wrap my head around the very last sentence without doubting my approach. Here's the question: Tony’s home has a ...
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1answer
57 views

Finding all possible values of $\alpha$ such that there is a likelihood test of size exactly $\alpha$

Let $X\sim \mathrm{Bin}(2,\theta)$ and test $H_0:\theta =0.5$ against $H_1:\theta=0.75$. Find the possible values of $\alpha$ for which there is a likelihood ratio test of size exactly $\alpha$.
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2answers
151 views

Probability to starve

First of all, I'm sorry if I'll use some game related terms, but that's where the question that bugged me for the last week came from. Let's say, we have a mana pool of size $M$, and we can cast a ...
2
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2answers
396 views

Likelihood ratio test of $n$ iid $\operatorname{Poisson}(\theta)$ random variables.

Let $X_1, X_2, ..., X_n$ be iid random variables, each with a Poisson distribution with parameter $\theta$. Find the form of the likelihood ratio test of $H_0: \theta = 1$ against $H_1:\theta=1.21$. ...
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1answer
182 views

Random Walk probability game

I try to solve some exercises from olympiads and I have difficulties with this one: Consider a round table with 20 people. One of these players receive a book and chooses one of his neighbors and ...
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1answer
1k views

Finding Mean Value and Standard Deviation

The distribution of resistance for resistors of a certain type is known to be normal, with 10% of all resistors having a resistance exceeding 10.256 ohms and 5% having a resistance smaller than ...
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1answer
39 views

$E[P^\mathcal{G}H;F\cap G]=E[(P^\mathcal{G}F)(P^\mathcal{G}H);G]$

In a proof of Proposition 5.6 ("conditional independence, Doob", p. 87) Kallenberg (1997) makes the following move whose justification eludes me: $$E\left[P^\mathcal{G}H;F\cap ...
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0answers
49 views

Mean matching size

Suppose there is a simple bipartite graph $G(X,E,Y)$, where $|X|=n_1$, $|Y|=n_2$, $|E|=m$. The edges $E$ are chosen uniformly at random. The question is what is a mean value of the size of the ...
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2answers
339 views

Let $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. Find $EZ$.

Let $X,Y$ independent random variables with $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. I already showed that $F_Z$ of $Z$ suffices $F_Z(z)=F(z)^2$. Now I need to find $EZ$. Should I start like ...
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0answers
103 views

How to find the cdf of the minimum of two r.v's?

Let $I=min\{0,W+V-U\}$ where $W,V,U$ are r.v's. Find the CDF of $I$ ?
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0answers
45 views

Second-order expected value given threshold

Suppose that $X_1, X_2, ..., X_n$ are uniform, independent random variables in $[0,1]$, and let $r\in[0,1]$. Suppose that the variables are ordered as $Y_1\geq Y_2\geq ...\geq Y_n$. Given that ...
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1answer
1k views

probability of hand with at least 2 kings

A hand H of 5 cards is chosen randomly from a standard deck of 52. Let E1 be the event that H has at least one King and let E2 be the event that H has at least 2 Kings. What is the conditional ...
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1answer
72 views

Binomial Random Variables: $P(X=k)$ as $k$ goes from $0$ to $n$

If $X$ is a binomial random variable with parameters $n$ and $p$, where $0 < p < 1$, show that As $k$ goes from $0$ to $n$, $P(X = k)$ first increases and then decreases, reaching ...
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2answers
213 views

biased coin flip. expected sequence

A biased coin $C$ has $\Pr(H) = a$ and $\Pr(T) = 1-a$. The coin $C$ is flipped $n$ times. What is the expected number of times that the consecutive sequence $HXH$ occurs where $X$ can be either $H$ or ...
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1answer
78 views

Joint distribution

This might be a trivial question. Let $X_{1}$ be a random variable and let $X_{2}$ be a random variable with the same probability distribution as the random variable $Y_{2}$. Question: Can I ...
2
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1answer
48 views

selecting marbles

An urn contains $r$ Red and $b$ Blue marbles. A fair coin is flipped. If the flip is Heads then $h$ Red marbles are added to the urn. If the flip is Tails then $t$ Blue marbles are add to the urn. Now ...
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1answer
43 views

Probability and Variance question

Can you help me solve this problem? Problem: You are playing a game, The game has 6 circles (O O O O O O) every time you play for 1 coin, each circle has a chance to be linked to the next one ...
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3answers
67 views

Verify my solutions to counting problems

I'm pretty sure I know the answers to these problems, but still want to double check. How many different ways are there of arranging all the letters of the string CALCULUSBOOK? Solution: $12!$ since ...
3
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2answers
57 views

Is the set $O:= \{(x,y) \in X \times X: d(x,y) < r \}$ Borel measurable?

Let $(X,d)$ a metric space and $\mathcal{X}$ its Borel sigma-algebra, i.e. the sigma-algebra generated by the open sets of $X$. Is the set $O:= \{(x,y) \in X \times X: d(x,y) < r \}$ $\mathcal{X} ...
3
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1answer
125 views

Definition of conditional expectation/independence

Conditional probability and conditional independence are unique almost surely, but relative to what: the conditioning field or the underlying field? More precisely, consider the case of conditional ...
3
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1answer
224 views

Selecting books from a shelf

A shelf contains 24 books. How many ways can 6 books be selected from these 24 with the restriction that no two selected books can be adjacent? So first we want to divide by 2 to fulfill the adjacent ...
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1answer
128 views

Probability of selecting jellybeans

8 red and 9 blue jellybeans are distributed randomly to 4 students. What is the probability that each student got at least one jellybean of each color? I am getting $\binom{7}{3} \binom{8}{3} / ...
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1answer
226 views

Expected number of coins taken by a pirate (problem with rounding)

We have N number of coins in a chest Two pirates are in a queue to take coins of the chest. When we draw some coins, the probabilities are all equal, so ${1,2,3....,k}$ all have the same probability ...
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1answer
707 views

Obscure Probability Question

Suppose that blood chloride concentration (mmol/L) has a normal distribution with mean 104 and standard deviation 5 (information in the article “Mathematical Model of Chloride Concentration in ...
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66 views

If $x \sim U(Z_n^*)$ then $x^2 \pmod n\sim U(QR_n)$?

Define: $Z_n^*=\{x \in Z_n | \operatorname{gcd}(x,n)=1\}$ $QR_n=\{x \in Z_n | \exists r \in Z_n \; s.t. \; r^2 =x\}$ How can I show that $x \sim U(Z_n^*) \implies x^2 \pmod n \sim U(QR_n)$? Thank ...