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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3
votes
2answers
45 views

Calculate the probability in winning tennis game.

A tennis tournament has $2n$ participants, $n$ Swedes and $n$ Norwegians. First, $n$ people are chosen at random from the $2n$ (with no regard to nationality) and then paired randomly with the ...
1
vote
0answers
25 views

Expectation of the fill distance of $N$ random points in $[0,1]^s$

Let $x_1,\ldots,x_N$ be uniformly distributed points in $[0,1]^s, s \in \mathbb{N}$. What can be said about $$ \mathbb{E} \left(\sup_{x \in [0,1]^s} \min_{i \in \{1,..N\}} \|x - x_i\|_2 \right) $$ ...
0
votes
1answer
41 views

Probability Average amount of rolls

I have a question regarding probability. I'll start by saying I've never taken a statistics or other similar course and was trying to work out this for a game. On average how many attempts will it ...
0
votes
2answers
26 views

Finding the mean of a truncated poisson pdf

Consider truncated pdf $y=1,2,3...$ $$f(y,\theta)=\frac{\theta^y e^{-\theta}}{y! (1-e^{-\theta})}$$ I'm required to find expected information (likelihood), but in short I need to find $E(\bar y)=E(\...
0
votes
0answers
17 views

Assigning weight to different factors to arrive at probability

Developing a report to determine probability of an outcome for a user. Looking to find the percentage chance that this probably will happen. Have about 10 factors that each have their own weight of ...
0
votes
1answer
36 views

Computing this conditional expectation?

Consider two random variables $X_1$,$X_2$ iid from a cdf $G$ on $[0,b]$. Consider also some concave function $f(.)$ with inverse $\phi(.)$ over $[0,b]$ such that $f(x)\leq x$, $\forall x\in [0,b]$. ...
0
votes
1answer
42 views

Find pdf given moments (well first by finding mgf!)

Suppose a continuous random variable has odd moments = zero and even moments as follows $$E[X^{2n}] = \frac{(2n)!}{2^n n!}$$ Then the mgf is, by Maclaurin series expansion where we can switch sum ...
0
votes
1answer
22 views

Distribution of the ratio of exponential r.vs with additive constant

I'm interested in the distribution of $\frac{X}{c+X+Y}$, where $X$ and $Y$ follows the exponential distribution with rate parameter $\lambda$, and $c$ is a constant. I know that without the constant $...
1
vote
0answers
27 views

Random walk with drift

Let $X_1,X_2,...$ be independent and identically distributed $\mathbb{Z}-$valued bounded random variables with mean $a=\mathbf{E}[X_1]$, and let $S_n = X_1+\cdots+ X_n$ be the associated random walk. ...
0
votes
0answers
17 views

Expected value of the exponential of a Geometric Brownian motion

I am trying to compute the following expectation: $$ E[ \exp (A_T)], $$ where $A_T = - C \int_{0}^{T} \exp( 2 \alpha W_t - \alpha^2 t) dt $, with $C$ and $\alpha$ positive constants, $W_t$ a standard ...
1
vote
0answers
60 views

Basic query related to conditional expectation

I have two variables $X$ and $Y$. I want to find the following probability $$P(X>a(Y+c),X+Y>d)$$ where $a>0,c>0,d>0$. To find the solution I have done following steps $$E_Y[P(X>a(Y+...
0
votes
0answers
15 views

Predictable stopping time

Assume an increasing rightcontinuous $(X_t)_{t\geq 0}$ has the compensator $(A_t)_{t\geq 0}$. As saz pointed out, we want to assume that $A_t$ is continuous. Define the stopping time $\tau_s:=\inf\{t\...
-2
votes
1answer
19 views

conditional distribution of minimum of a constant and a random variable [closed]

Let $S$ be an Exponentially distributed random variable with parameter $\lambda$. We define $T=\min(S,L)$ where $L$ is a fixed constant. In this case, what is the conditional distribution $P(t\mid s)$?...
0
votes
1answer
17 views

Sample space in probability computing

This is a simple example of probability computing. There are $n$ white balls and one black ball in a box. Take the balls one by one out of the box until the black ball appears. Let $X$ denotes the ...
1
vote
0answers
38 views

Is it possible to be a frequentist and a subjectivist at the same time?

I'm trying to understand the differences between (1) Bayesian vs frequentist; and (2) subjectivist vs objectivist. So far my understanding (correct me if I'm wrong) is that: (1) Bayesian vs ...
0
votes
2answers
29 views

Fine E(4X^2+4X+1)

So I have the following tables $$ \left[ \begin{array}{c|ccc} x&-3&6&9\\ f(x)&\frac{1}{6}&\frac{1}{2}&\frac{1}{3} \end{array} \right] $$ I am tasked to ...
0
votes
1answer
27 views

Quality of a signal

Let's consider a signal y(t) that is an output of a linear transformation with the addition of noise. $$y ( t ) = Ax ( t − \theta) + m ( t )$$ The hypothesis on the signals x(t), y(t) is that they ...
0
votes
1answer
43 views

Can we use a symmetry argument instead of integration in BASIC probability?

Suppose $H$ is a random variable with pdf $f_H(h)$. Let $X$ and $Y$ be random variables with joint pdf $$f_{X,Y} = f_H(x) f_H(y)$$ Prove $$P(X \ge Y) = 1/2$$ Is it possible to ...
0
votes
0answers
20 views

Finding best strategies in a problem about traffic lights

A problem came up in a course of a conversation with my friend. Suppose we have a street with several traffic lights, placed equidistantly one from another. Time of them being green $t_g$ and time of ...
0
votes
0answers
38 views

Prove that if $X$ and $Y$ are independent, then $h(X)$ and $g(Y)$ are independent in BASIC probability — can we use double integration?

In advanced probability we can just say: \begin{align} & P(h(X) \in A, g(Y) \in B) \\[6pt] = {} & P(X \in h^{-1}(A), Y \in g^{-1}(B)) \\[6pt] = {} & P(X \in h^{-1}(A)) P(Y \in g^{-1}(B)) \...
2
votes
2answers
26 views

How do you find the distribution of Y?

the problem says that $X$~$N(μ,σ^2)$ and $ Y=1.2X+3.8$ And you need to find the distribution of Y. I tried to apply the transformation of random variables. $Fy(y)=p(Y<y)=p(1.2X+3.8<y)=p(x<(y-...
0
votes
2answers
45 views

Conditional probability involving poker cards

In poker game card, $ 52 $ cards are distributed equally among $ 4 $ players $ A, B, C, $ and $ D. $ If $ A $ and $ B $ have a total of $ 8 $ spades, what is the probability that $ C $ has $ 3 $ of ...
6
votes
2answers
169 views

Fun with combinatorics and 80 business customers

In business with 80 workers, 7 of them are angry. If the business leader visits and picks 12 randomly, what is the probability of picking 12 where exactly 1 is angry? (7/80)(73/79)(72/78)(71/77)(70/...
2
votes
1answer
71 views

Finding Size-Bias Distributions

For a RV $W$ with mean $\mu$, let $W^*$ denote the $W$-size biased distribution (so that $EG(W^*)=\frac{E(WG(W))}{\mu}$ for all functions $G$ for which the expectations exist). I would like to learn ...
1
vote
1answer
44 views

Roulette and Discrete Distribution

A roulette wheel has 38 numbers. Eighteen of the numbers are black, eighteen are red, and two are green. When the wheel is spun, the ball is equally likely to land on any of the 38 numbers. Each spin ...
1
vote
0answers
23 views

Statistical test for convergence in distribution

Please forgive my total ignorance on all things statistical. I have some sequence of distributions (actual data, a collection of real numbers for each positive integer, which don't appear to converge ...
5
votes
2answers
66 views

What's the probability of getting $5$ different numbers but not any $6$ when throwing $5$ dice?

I have $5$ dice, I throw them at once. What is the probability of getting $5$ unique numbers, i.e., $1\ \ \&\ \ 2\ \ \&\ \ 3\ \ \&\ \ 4\ \ \&\ \ 5$ in any order BUT NOT any $6$?...
1
vote
1answer
57 views

Probability of $|H-T|$ in 10,000 coin tosses

If a fair coin is thrown $10,000$ times. Using the binomial convergence to normal,find $P|H-T|\le 80$ My intuition say that mean is 0.But I am not able to proceed further.
0
votes
0answers
37 views

Using Chebyshev inequality to find limit theorem probability

Limit theorem states that for $x\rightarrow a$, if $f(x) = L$ then for $0<|x-a|<\delta$, there is $|f(x)-L| < \epsilon$. The Chebyshev's inequality states that: $P(|x-a|\leq ky)$ is less ...
0
votes
1answer
21 views

Convergence of random variable 5

If $Q < \frac{n}{m^2} X_n$ where $X_n$ is a sequence of random variables, $X_n \xrightarrow{a.s}1$, $0\leq Q \leq1$, $m=\omega(\sqrt{n})$ (The $\omega$ denotes the order, see here). Then, how can ...
0
votes
2answers
43 views

$\mathbb{E}[|X|^n] < +\infty \implies \mathbb{E}[|X|^k] < +\infty, k \leq n$

Show that if $\mathbb{E}[|X|^n] < +\infty$, then $\mathbb{E}[|X|^k] < +\infty, \forall k \leq n$. I guess I have to apply Hölder Inequality, but I was not able to find out what $p$ and $q$ are ...
1
vote
1answer
31 views

Transformation of density and $W=(X+Y+Z)^2$

I want to solve this exercise with the transformation formula, what did I do wrong in my solution?: Let $X,Y,Z$ be independent random variables with uniform distribution on [0,1]. What's the ...
0
votes
1answer
19 views

Identity of Running maximum

let $X_t$ denotes a arithmetic Brownian motion process. I am wondering if the following identity is true ? $$ \mathrm{P}\left[\sup_{0 \le s \le t} \mathrm{e}^{X_t} < x\right] = \mathrm{P}\left[\...
1
vote
1answer
49 views

Why the probability is $0$ but possible

We want to take a random number from natural numbers how much is the probability that,the number be $1$? When we want to say the probability we say it is $0$ but we say zero for impossible things but ...
0
votes
0answers
27 views

Forecasting Query on probabilities

many Thanks if you can help with this query. Player A and Player B are due to play a darts match. Player A has a record of scoring between 80 and 140 in each of his last 35 rounds of throws. Best ...
-5
votes
1answer
31 views

Stats question bionomial distribution. [closed]

It is known that 47% of students at a large university are male. If we take a random sample of 200 students at the university, what is the approximate probability that less than half of them are male? ...
1
vote
0answers
33 views

The average number of transitions to a particular state from only two particular states.

I know that in a Markov chain, $$\mathbf{N} = (\mathbf{I} - \mathbf{Q})^{-1}$$ gives a matrix to calculate the expected number of times before absorption that a particular [transient] state is visited,...
0
votes
1answer
26 views

Interesection of ranges r.v. and max of r.v.

I have the following question, let $X_1,...,X_n$ be some discrete random variables, and let $Y$ and $Y'$ be two geometric random variables that are not necessarily independent of $X_i$. let $p_Y \geq ...
1
vote
1answer
25 views

How to find normal distribution that has a quadratic?

Let $X$ be a normal random variable with mean 1 and variance 4. Find $P(X^2 − 2X ≤ 8)$. (Answer key .86) My attempt $$P(X^2-2X\le 8)=P((X+2)(X-4)\le 0)$$ and this is where I am lost. I did the ...
1
vote
2answers
37 views

compute marginal

I have tried to solve this exercise Let $X$ and $Y$ be random variables with joint probability density function given by: $f(x,y)=\frac{1}{8}(x^2-y^2)e^{-x}$ if $x>0$, $|y|<x$ Calculate $E(X\...
0
votes
1answer
18 views

Probability Density Function Example Help

I am reading through the text, "Introduction to Probability Theory with Contemporary Applications" by Lester Helms. I am stuck on the attached example. I understand how the author obtained the joint ...
0
votes
1answer
37 views

Find a 95% upper confidence bound for lognormal dispersion

Let $X_1,\ldots,X_n$ be a sample from a population $X$. Assume that $X$ has a $lognormal$ distribution $(2,\sigma^{2})$, with $\sigma^{2}$ an unknow parameter. Find, for the variance of the population,...
2
votes
3answers
91 views

Combinatorics on the word Abracadabra

How many different 'words' can be created using all the characters of 'ABRACADABRA'? In how many of the 'words' that there are no identical characters one next to the other? So, For the first part, ...
0
votes
1answer
55 views

how are these probability rules derived? [closed]

I was reading a note and got to the following equations. Not sure how they are derived. $$\Pr(\overline{E_t} \cap · · · \cap E_0) = \Pr(E_{t−1} \cap · · · \cap E_0)−\Pr(E_t \cap · · · \cap E_0)$$ and $...
0
votes
0answers
18 views

Is Site Percolation with Bernoulli variables i.i.d. independent and identically distributed?

I cannot understand the identically distributed part in the i.i.d assumption. Consider a site percolation where each event is a Bernoulli variable. Does this mean ...
2
votes
0answers
44 views

Almost sure convergence (correctness of an argument)

Is this statement correct? If $X_n \xrightarrow{a.s} c$, where $X_n$ is a sequence of random variables and $c$ is a constant, then we can conclude that since almost sure convergence implies on ...
1
vote
0answers
18 views

Stratified Sampling $E(E(X \mid \mu, \sigma))$

Let $X$, $\mu$ and $\sigma$ be random variables. I want to estimate $E(X)$ using Monte Carlo. I am able to sample from, and know in closed-form, both the conditional distribution of $X \mid (\mu, \...
2
votes
0answers
20 views

Means and Covariances of powers of a normal distribution

Let $X$ be a normally distributed random variable, with mean $\mu$ and variance $\sigma^2$. Consider a random vector $$V = \left[ X^n, X^{n-1}, \dots, X^2, X, 1 \right]^T $$ What is the expected ...
4
votes
2answers
966 views

Monkey typing on 29 letter keyboard.

This monkey is driving me a little crazy. I think he should get fired - it's not nice. Here is the information. A monkey is typing on a 29 letter keyboard. He is writing a word that is 5 letters ...
3
votes
1answer
29 views

Convergence sequence of mean implies convergence in mean / weakly consistence of subsequence of regression function estimates

Let $(X_n)$ be a sequence of positive random variables. Suppose that the limit of expectation of this sequence $\lim_{n\rightarrow\infty}\mathbb{E}[X_n]=0$. This imply that $(X_n)$ converges to zero ...