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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1answer
132 views

Waiting time in an immigration-birth process

Could someone please verify that none of the four given choices are correct? Isn't the correct answer $$\frac 1{(\lambda + 4\beta)^2} + \frac 1{(\lambda + 5\beta)^2} + ... +\frac 1{(\lambda + ...
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4answers
71 views

Simplifying this expression involving binomial coefficients.

\begin{align*} f(y) &=\left(\frac\delta{1-\delta}\right)^y(1-p)^n\sum_{u=0}^{n-y}\binom{u+y}y\binom n{u+y}\alpha^{u+y}\\ ...
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2answers
72 views

Expected value for people which doesn't have neighbors on a bar branch

3 people get into a bar and sit randomly on a bench with n-sits. Find the a.Variance of people which don't have neighbors where $n=5$ b. Expected value for people who don't have ...
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1answer
38 views

Can I use matrix algebra to solve this probability?

From another discussion forum: One throws dice $n$ times. What is the probability that during $n-1$ times one has got five different values and gets the missing sixth value on his last throw? I ...
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1answer
303 views

Finding Probability of P(|X-Y| ≤ 0.5)?

The joint density of X and Y is given by f(x,y) = (x + y); 0 < x,y < 1 = 0; otherwise Find ...
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1answer
375 views

Finding the unconditional distribution

I found two similar questions. One has a good answer What is the distribution of an unconditioned random variable knowing the conditional distribution? . I have a similar problem that I think should ...
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1answer
33 views

Combinatorial probability: how often an element appears in a random set of $n$ elements

A friend of mine asked me to help him with the following problem: basic probability calculus on a discrete set, but I need some confirmation whether my combinatorial considerations are OK. Let ...
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2answers
249 views

conditional probability question from sheldon ross

In any given year a male automobile policyholder will make a claim with probability $p_{m}$, and a female policyholder will make a claim with probability $p_{f}$, where $p_{f} \neq p_{m}$. The ...
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1answer
72 views

Misclassification Error probablity

Let ρ be a probability distribution on $Z = X \times Y$ and $(X,Y)$ be the corresponding random variable . The missclassifications error for a $f:X\to Y $ is defined to be the probability of the ...
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1answer
48 views

Optimization of expected value of random variable

Z is random variable assuming integer values n1,n2,..nM with ni>0 for every i=1,2..,M PMF qof z p1,p2,...,pM Problem: Maximize ...
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2answers
1k views

Expected Value of Number of Tails Minus Number of Heads

I have the following problem where, given $X_n$ is a random variable that equals the number of tails minus the number of heads when n fair coins are flipped, what is the expected value of $X_n$? I am ...
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0answers
50 views

probability of a collision on a n bit secure hash.

I have a homework question that I need confirmation on. Wikipedia states that a hash of n bits can be broken in $2^{n/2}$. Meaning that $2^{n/2}$ of hashes is expected to be computed in order to ...
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0answers
259 views

Probability answer verification

Of a group of patients having injuries, 28% visit both a physical therapist and a chiropractor and 8% visit neither. Say that the probability of a visiting a physical therapist exceeds the probability ...
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2answers
47 views

Discrepancy When Calculating Variance

I have the following problem, asking what is the variance of the number of successes when $n$ independent Bernoulli trials are performed, where, on each trial, $p$ is the probability of success and ...
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1answer
445 views

Probability of events using geometric series

$P(\{n\}) = (1/2)^n$ $A = {n : 1 <= n <= 10}$ $B = {n : 1 <= n <= 20}$ $C = {n : 11 <= n <= 20}$ Find: $P(A), P(B), P(A \cup B), P(A \cap B), P(C)$ and $P(B')$ I need to use the ...
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1answer
1k views

Probability of finding P(X=k)?

A factory produces 10 glass containers daily. It may be assumed that there is a constant probability p=0.1 of producing a defective container. Before these containers are stored they are inspected and ...
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0answers
80 views

multivariate non-negative distributions

I need a multivariate distribution defined over any vector (not special matrices as in case of Wishart distribution etc.) whose elements are non-negative (must include 0 as well). There is no other ...
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0answers
78 views

Expected value, correlation, and indepence.

I need help with a problem. Supposed x, y, and z are events in F (algebra of sets) in a probability space (universal set, F (algebra of sets), P). Define two random variables: a(omega) = ...
2
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1answer
161 views

Using Central Limit Theorem when we NON-IID sample

I'm trying to solve a CLT question and I've got some issues. I appreciate if you could help me on that. Consider the question below: $\epsilon_i$ 's are iid random variables with finite mean and ...
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2answers
64 views

What do I need to study to do this Gaussian question?

I'm taking a probabilistic machine learning course and need to understand some background mathematics, including the following question: Let $x$ be a Gaussian random variable with mean $μ$ and ...
1
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1answer
42 views

Probability for not finding a product

A product is exist in $\frac{1}{4}$ of chain stores. One decided looking for the product in not more than 5 stores. Defining $X$ to be number of searched stores, find: a.Distribution of $X$ ...
1
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1answer
53 views

Maximize probability of working with the smartest person in a group of people.

My biggest regret in choosing courses in university was choosing statistics over probability. Hence, I have a problem approaching this question, and fear my skills in probability are insufficient. ...
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2answers
106 views

95% confidence interval on demand

A company selling oil operates a storage tank to serve 10 customer locations. The monthly oil demand at each customer location is normally distributed with mean 50 million gallons and standard ...
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0answers
54 views

How to learn mixture Gaussian with inequality constraint of component variances

Let $f_1(x)$,…,$f_n(x)$ be Gaussian density functions with different parameters, $\mu_i$ and $\sigma_i$ are the parameters (mean and variance) of the Gaussian component i, and $w_1,\ldots,w_n$ be real ...
2
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1answer
117 views

Calculating the probability of following event involving Brownian motion

I have a big time trouble in evaluating the following probability. It is related to brownian motion and measure, so I am asking experts from both fields for help! Denote $B_t$, $t\in [0, T]$ be ...
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1answer
377 views

n indistinguishable balls are to be arranged in N distinguishable boxes

Suppose that n indistinguishable balls are to be arranged in N distinguishable boxes so that each distinguishable arrangement is equally likely. If $n \geq N$, show that the probability no box will be ...
2
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2answers
60 views

Expected Cost while gambling

It cost \$35 to gamble for the golden idol. There is a 1% chance that you'll get it, and a 99% chance that you'll get a common household item. Each gambler can own at most one golden idol, and ...
0
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1answer
151 views

If random variable $(X \, | \, Y=y) \sim \mathcal{B}(y,p)$ then $\mathbb{E}[X\color{red}{^n} \, | \, Y=y] = yp$?

Refering to equation marked (2): I am aware if $(X \, | \, Y=y) \sim \mathcal{B}(y,p)$ then $\mathbb{E}[X \, | \, Y=y] = yp$? but if its $X^n$ where $n$ is a constant? Does it still hold? It ...
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1answer
174 views

E(XY) expectation product

Could someone give an example on when and why you would want to multiply two random variables? What is the reason? For example, what does the expectation E(XY) mean in intuitive terms? Thanks in ...
2
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2answers
108 views

Beta distribution quesions

Just a simple beta distribution question just to be sure that I understand it. Say we do experiments, and we expect a proportion $\theta$ of people having a specific property (which means $\theta ...
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1answer
2k views

Probability density of Continuous uniform distribution over the unit circle

If we want to chose a point $(x,y)$ uniformly at random from a unit circle in a plane, why is the joint probability density of the random variable $f(x,y) = \frac{1}{\pi}$ for $x^2+y^2\leq1$? The ...
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1answer
501 views

Calculations for a random variable given density function

Studying for a statistics course an stumped on how to go bout solving a problem. The following is a problem I have solved - the problem I'm having trouble with is related to this one: ...
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0answers
133 views

Limiting behavior of an integral involving incomplete Gamma function

I am wondering about the limiting behavior as $k\rightarrow\infty$ of the following integral: $$I(k)=\frac{2^{-k/2}}{\Gamma(k/2)}\int_{f(k)}^\infty ...
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1answer
49 views

Marginal distribution of $P$

Joint distribution of $P$ & $Q$ is $$f_{P,Q}(p,q)=\frac{1}{2\sqrt{(2\pi)}\sigma}\exp[-\frac{1}{2}{(\frac{\frac{p+q}{2}-\mu}{\sigma})^2}] \times\theta\exp[-\theta(\frac{p-q}{2})],\quad ...
2
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2answers
113 views

Expression for Expectation of expectation

Suppose we have two random variables: $X$ is a continuous r.v.; $Y$ is a discrete r.v. taking values $0$ and $1$. Is the following expression true ? $E[(E[X|Y])^{2}]= [(E[X|Y=1])^{2}]\times P(Y=1)+ ...
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1answer
72 views

Expectation of $[Y g(X)]$

Firstly, how do I interpret $\mathbb{E}[g(X)]$. I understand $\mathbb{E}[X]$ is like the most likely outcome of a set of experiments (loosely speaking at least - not really a very maths person)? But ...
0
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1answer
26 views

Simplification Formula (conditional expec)

Is there any known formula linking $E[D\mid (S=0) \cap (X=j)]$ and $E[D\mid (S=0)]$ given that S and X are 2 indepedent random variables? Thanks
2
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1answer
88 views

A variance-mixture model

So I've tried to make a probability distribution which has a tunable degree of kurtosis and which becomes Gaussian if the control-parameter goes to 0. Now Levy-distributions are out of the question, ...
2
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1answer
19 views

A question about distributions/densities

Given two random variables $X,Y$ how to show that $P(X\leq Y+x)=\int F_X(y+x)f_Y(y)dy$? I know that $f_Y(y) = \int f_{XY}(x,y)dx$, but have no idea how to go with the previous equation.
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0answers
89 views

What distribution describes the number of balls in bins of limited size

Another balls into bins question... I have n indistinguishable balls to distribute into k bins which can hold no more than l balls. What probability density function describes the number of balls in ...
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1answer
146 views

Problem on conditional probability.

A small brewery has three bottling machines. Machine $A$ produces $40\%$ of all the bottles, machines $B$ and $C$ produce $30\%$ each. Five percent of bottles filled by $A$, four percent of bottles ...
3
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3answers
126 views

If $X\sim \exp(\lambda)$ and $Y\sim \exp(\mu)$ then $P(X\leq Y)=\frac{\lambda}{\lambda+\mu}$. Is there an intuitive interpretation for this fact?

I can verify this via double integrals, but I'm wondering if this can be put in the context of a Poisson process or something to give it an obvious meaning. I can't think of exactly how it would work. ...
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3answers
532 views

Probability a die will come up 6 at least twicer in twelve rolls

What is the probability of rolling at least two 6's on twelve rolls of a fair 6-sided die. I am using the complement to solve the question $$1-\frac {5^{12}+(_{12}C_ 1)5^{11}}{6^{12}}$$ $5^{12}$ is ...
2
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2answers
114 views

Why is this standard deviation $20$?

If I have two random variables $X_1$ and $X_2$ with $X_1\sim N(520,10)$ and $X_2\sim N(500,10)$, and $X_1$, $X_2$ are both speeds of airplanes where the first one is 10 km ahead of the second one. I'm ...
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1answer
237 views

Application of Bayes' theorem - probability problem Suppose that the reliability of a HIV test is specified as follows : Of people having HIV, …

Bayes' Theorem States : *If $E_1,E_2,....E_n$ are n non empty evnents which constitute a partition of sample space S, ie.e. $E_1,E_2,....E_n$ are pairwise disjoint and $E_1 \cup E_2 ......\cup E_n$ = ...
2
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1answer
172 views

A problem on binary number

Let $B_n$ be $n$-bit binary number. Each bit could be either 0 or 1 with equal probability and mutually independent. Let $b_i$ be the $i^{th}$ bit of $B_n$. Let $Z_{ij}$ be the decimal value of the ...
0
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1answer
283 views

Are $X$ and $Y$ independent or uncorrelated? [closed]

let $Z$ be a uniformly distributed random variable over the range $[-1,1]$ let $X=Z$ and $Y=Z^2$ be random variables. a) Are $X$ and $Y$ independent? b) Are $X$ and $Y$ uncorrelated?
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2answers
35 views

Mean of probability distribution function.

The current chapter I am working on is continuous random variables. I know that the mean value of a continuous random variable is: $$ E[X] =\int_{-\infty}^{\infty} xf(x) dx $$ That being said, my ...
0
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1answer
29 views

Does this have a limit?

Does this have a limit as $n \to \infty$ : ${\alpha}^{2(-1)^{n + 1}n}{\beta}^{(-1)^{n}(2n+1)}$ and $0< \alpha, \beta < 1$.
2
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1answer
208 views

Reciprocal of a Binomial

I'm wondering if there is any formula to do this. Suppose $B$~$B(N,p)$, and hence we have $E[B]=Np,D[B]=Np(1-p)$. I'm just wondering how to do $E[\frac{1}{c+B}]$,for some $c>0$? Thanks.