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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0answers
6 views

Propability theory Limsup excercise

Let $(\Omega,F,P)$ be a propability space and $(A_n)$, $n\in\mathbb{N}$, belonging in $F$ for every $n$ and independent. Suppose $P(A_n)<1$ for every $n$ and $$P\left(\bigcup_{n=1}^{\infty} ...
-1
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0answers
15 views

Cube of Brownian motion [on hold]

Find all $H_t$ so that: $B^3_T = \int_0 ^T H_t dB_t$ $\int_0^T B^3_tdB_t = \int_0^T H_t dB_t$ where $B_t$ is a Brownian motion.
2
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0answers
27 views

Is my method of working fine?

Suppose a point $X$ is selected at random from a line segment $AB$ of length $l$ and midpoint $O$. Find the probability that $AX,BX$ and $AO$ form a triangle. My method and working is: Case ...
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0answers
7 views

Using Conditional Jensen inequality proof the following

$X_1,X_2,\ldots,X_n$ are i.i.d. random variables, $X_1>0$, $E[X_1]=\mu$, $E[X_1^k]<\infty$ for $1<k \leq2$. Proof: $$ E\left[\left(\frac{1}{n}\sum_{i=1}^nX_i\right)^k\right]\leq ...
-2
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0answers
22 views

conditional probability proof 3 varables [on hold]

Suppose that $\mathcal a$ ,$\mathcal b$ and $\mathcal c$ are dependent variables. $$\mathbb P(a \mid b) = \sum \mathbb P(a \mid b,c) \ \mathbb P(c \mid b)$$ can anyone explain it how we get it?
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2answers
17 views

Rolling dice probability by solving inequlity

I was trying to solve a problem where I have to find the probability of the sum of $\mathcal 3$ rolls of a die being less than or equal to $\mathcal 9$. In order to solve the problem I try first to ...
0
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0answers
11 views

Probability of a sequence of urn draws having some pair of draws with a minium number of “matches”?

I have $U$ urns. Each urn contains some sequentially numbered balls (not necessarily the same count between urns) $1, 2, 3,... N_u$. I draw one ball from each urn $1, 2, 3,...U$ in turn, and note ...
0
votes
0answers
18 views

How to use joint probability density to check for independent events?

Suppose that the joint PDF of $X$ and $Y$ is as follows: $$ f(x) = \begin{cases} 24xy & \text {$x \geq 0, y \geq 0, x+y \leq 1$}\\ 0 & \text {otherwise ...
1
vote
1answer
15 views

Covariance of $2$ variables

I am given two random variables $X$ and $Y$. I am also given that $\mathbb{E}(Y|X)=\mathbb{E}(Y)=\mu_y$ and $\mathbb{E}(X)=\mu_x$. So if I need to calculate the covariance of $X$ and $Y$, ...
4
votes
1answer
29 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...
1
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1answer
16 views

Question about asymmetry of chi-square distribution

Let $X_1,\dots,X_n$ be a set of i.i.d. chi-square random variables with $k$ degrees of freedom. Consider the statistic $\arg\max_i\{|X_i/k - 1|\}=X_{\alpha}$. I wonder about the probability that ...
0
votes
2answers
15 views

Multinomial Coefficients Dice Problem

If 7 balanced dice, are rolled, what is the probability that each of the 6 different numbers will appear at least once? My attempt: $p=\frac{7!}{2!6^6}$ So if 6 different numbers need to appear, ...
2
votes
3answers
32 views

Probability of balls in boxes

If $12$ balls are thrown at random into $20$ boxes, what is the probability that no box will receive more than $1$ ball? So my book says the answer is: $\displaystyle \frac{20!}{8!20^{12}}$ However ...
1
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1answer
28 views

Calculate the mean, the median and the quartiles.

Let $D=\{(x,y):x>0,x^2+y^2<1\}$ and let $(X,Y)$ be the random variable with the density: $$f(x,y)=\frac{2}{\pi}1_{D}(x,y).$$ Let $Z=\frac{Y}{X}$. Calculate the mean, the median and the first and ...
1
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1answer
25 views

Probability and Statistics (Normal Distribution)

Having trouble with the last part of this question. Not sure how the man would divide his pile of vouchers? It seems that you could interpret this question in a lot of ways. Any tips would be ...
0
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0answers
7 views

Obtaining the Log-logistic distribution from a truncated logistic distribution

Let $$f(x) = \frac{e^x}{(1+e^x)^2}~,~ -\infty \lt x \lt \infty~~~~~(1)$$ be the standard logistic pdf of a random variable $X$. Then one can obtain the pdf of the log-logistic distribution via the ...
-1
votes
0answers
20 views

Martingale Ideas in Elementary Probability [on hold]

There is a non-symmetric version of the probability model in which the probability of success on each trial is "$p$" and the probability of failure on each trial is "$q$" and $p+q=1$. The probability ...
0
votes
1answer
26 views

Calculate the probability given by three random variables

Let $X_1,X_2,X_3$ be IID random variables, each with the density $$f(x)=x e^{-x}\cdot 1_{(0,\infty)}(x).$$ Calculate $P(X_1+X_2+X_3>4,X_1+X_2<4)$.
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0answers
10 views

Question on Standard Brownian Motion [on hold]

What is the following probability $P [ W(2) >0 \ \text{and}\ \ W(1) <0]$?
2
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2answers
24 views

Convergence in law of sample means of random variable

Let $\{X_n | n \in \mathbb{N} \}$ be a sequence of independent identically distributed random variables with density function: $$f_X(x) = e^{\theta - x}I_{(\theta, \infty)}(x)$$ with $\theta > ...
0
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0answers
16 views

Show that for any geometric random variable $X$ and parameter $p, \mathrm{Pr}(X < t) = 1 − p^t$. [on hold]

How to prove the above stated equation? I tried the following : Pr⁡(X(i=1)^(t-1)▒〖Pr⁡(X=i)〗 =∑(i=1)^(t-1)▒〖p(1-p)i-1〗 =1-(1-p)t-1
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0answers
29 views

Conditional Probability Proof for three events

For any 3 events $X, Y$ and $Z$ where $\Pr Z) > 0$, it is required to prove that $$\Pr ((X \cup Y) \mid Z ) = \Pr(X\mid Z) +\Pr(Y\mid Z) - \Pr ((X \cap Y) \mid Z)$$ I am not able to prove ...
0
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1answer
41 views

How to prove the folowing theorem in probablity? [duplicate]

Show that for any continuous random variable $X$ that takes only positive real values $\int_{0}^{\infty}\text{Pr}(X\geq x)dx=\mu$ where $\mu$ is the mean.
2
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1answer
23 views

Can two nodes in a Markov chain have transitions that don't total 1?

In all the Markov diagrams I see, the transitions from state A to B always total to one. Just one of many examples, this image ...
0
votes
1answer
31 views

Estimator for second moment for Poisson random variable

Let $X \sim Poiss(\lambda)$. As, $\displaystyle \sum_{i=1}^{N} X_i $ is sufficient statistic for both mean (and variance) of $Y$, so we can define the unbiased estimate for mean as , $ s=\frac{1}{N} ...
1
vote
1answer
25 views

Obtain the MGF of $Y$.

Let $X$ be a random variable whose probability density function is: $f_{X}(x)=e^{-x}$ Then, obtain the Moment Generating of function of $Y=1-e^{-X}$ What I did: We can find a bound for $y$ using ...
0
votes
1answer
25 views

Sigma-fields and probability

I'm unsure what this question asks of me. For (i) I have given a power set with 16 elements in terms of a,b,c and d. I don't understand what I need to do for (ii). I believe (iii) is fairly ...
3
votes
2answers
53 views

Probability of winning a tie-break in tennis?

The winner of a tennis tie break is the first to get to 7 points and lead by 2. Let $p$ be the probability of player 1 winning when serving, and let $m$ be the probabiliity of player 1 winning when ...
1
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1answer
23 views

If the probability of 3 events with non-zero probability equals the product of the individual probabilities, are they also pairwise independent?

Consider three events $A$, $B$, and $C$, none of which has a zero probability. If $A$, $B$, and $C$ satisfy $\Pr(A \cap B \cap C) = \Pr(A) \cdot \Pr(B) \cdot \Pr(C)$, does this imply that the three ...
2
votes
1answer
29 views

Probability of winning a game in tennis?

Suppose there is a tennis singles match, where Player A plays a single game against Player B. The probability that player A will win a single point is $x$, and thus $1-x$ is the probability that ...
1
vote
1answer
22 views

Sum of uniform random variables $U(0,1)$ and $U(0, a)$

The problem I have is: $X \sim U(0,1), Y \sim U(0,a)$ are independent random variables. Find the pdf of $X + Y$. I've got stuck in an integral-problem, and will show you what I've tried. Skip to the ...
0
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0answers
27 views

How to demonstrate the pdf of $P_{\sigma} (t)=\lambda_c e^{- \lambda_c t} / (1 - e^{- \lambda_c T})$

In $t_c$, there are $n$ expirations of $T$ and the remnant $\sigma$ seen from the above figure. Let the time $t_c$ forms the exponential distribution with parameter $\lambda_c$. How to demonstrate ...
0
votes
1answer
32 views

If three events are independent, are they also pairwise independent?

If three events A, B, and C satisfy $\Pr(A \cap B \cap C) = \Pr(A) \cdot \Pr(B) \cdot \Pr(C)$, does this imply that the three events also satisfy the following? $\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)$ ...
0
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0answers
19 views

Dominated Convergence Theorem.

Dominated Convergence Theorem "Suppose $X_{n}\rightarrow X$ a.s., and there is a random variable $Y$ with $E[Y]<\infty$ such that $|X_{n}|<Y$ for all $n$. Then $E[lim_{n \to ...
0
votes
1answer
6 views

CDF of the kinetic energy of a particle under uniform distribution

We are given that X~Uniform[2,3] and the kinetic energy is $T=\frac{1}{2X^2}$ I tried the following: $P(T\leq a) = P(\frac{1}{2X^2}\leq a) = P(-\sqrt{2a}\leq X \leq \sqrt{2a}) = ...
1
vote
1answer
35 views

A Seemingly Trivial but Computationally Complicated Probability Problem

Suppose $X,Y$ are independent $Uniform(-1,1)$ random variables. Determine the distribution of $Z=X-Y$. I do not really think I should add my work here because whatever I have tried until now, has ...
0
votes
1answer
13 views

Conditional Probability Problem With Cards

X = # of Aces Y= # of Kings $h(X$ | $y=2) = \frac{f(x,y)}{f_Y(y)}$ Need help with what to do next. Edit 1: This is the function I came up with: $ h(X$ | $2) = \frac{ \displaystyle \binom{4}{x} ...
1
vote
1answer
14 views

density of 2 bivariate gaussian random variables

$X_1$ and $X_2$ are bi-variate Gaussian with equal mean and variance. how do i find the density of & $y = A_1X_1 + B_1X_2$.? I think I should use correlation co-efficient here which i assume as ...
-2
votes
1answer
26 views

how do I find the expected payout? [on hold]

You can roll a dice three times. You will be given $X$ where $X$ is the highest roll you get. You can choose to stop rolling at any time (e.g., if you roll a $6$ on the first roll, you can stop). What ...
1
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1answer
23 views

expectation of uniformly distributed $n$ number of samples

I am trying to fine the expectation: $E((x_1+ x_2+ \cdots +x_n )^2)$ as a function of $n$ where all $x_1$ to $x_n$ have uniform distribution $U(0,1)$. I can do if there is only $x_1$ and $x_2$ but ...
0
votes
0answers
21 views

Limit definition of Sets.

Proposition 1.32 $X_{n}\xrightarrow{a.s.} X$ if and only if for any $\epsilon>0$ $P( | X_{n}- X |<\epsilon, \; \forall n\geq m )\rightarrow1$ $as$ $ m\rightarrow\infty$ Proof. Suppose first ...
2
votes
1answer
25 views

Find the covariance of $Y_1$ and $Y_2$

I had a statistics question I was hoping for help on: Let $Y_1$ and $Y_2$ be discrete random variables with join probability function: $$f(x,y) = \begin{cases} \dfrac{y_1 + 2y_2}{18} & \text{if ...
0
votes
1answer
22 views

ways to choose 4 people including two or more male from N people with 1/3 male 2/3 female

A sample of four people is randomly drawn from a population of N > 4 people.Assume that 1/3 of the total population is male, and 2/3 is female. (To simplify things, let’s assume that N is always ...
1
vote
0answers
23 views

Central Limit Theorem proof.

I am trying to understand the proof of the Central Limit Theorem in my book. However, I don't really understand what is going on. I know the proof is assuming that the moment generating functions of ...
-1
votes
0answers
23 views

How to prove stochastic dominance? [on hold]

Consider the set of constant vectors $p_i$ and $\tilde{p}_i$, such that $p_i \succeq \tilde{p}_i \succ \mathbb{0}\; \forall i$ (component wise inequality) and define: $M \triangleq ...
2
votes
1answer
33 views

A variation of a combination and a permutation, I think?

The scenario is that 6 people have the option of choosing 8 doors and we want to know each door a person goes through. I have four/five questions based on this. 1) How many different ways can 6 ...
0
votes
2answers
15 views

Probability of a word where 2 letters do not follow each other

I have seven letters, say A, B, C, D, E, E, G. I have figured out how many distinct possible combinations I can have as $7!/2!$. My question is, how many of these will have the two E's separated? I ...
0
votes
1answer
7 views

Relative frequency

In a given scenario where two fair dice are thrown: what is the probability of the second roll being higher than the first? I can think of two ways to resolve this problem; 1- listing the possible ...
0
votes
1answer
24 views

Doob-Kolmogorov Inequality

Denote by $(X(t),t\ge 0)$ a standard Brownian motion, i.e random variables with the following properties: $X(0)=0$. With probability 1, the function $t\mapsto X(t)$ is continuous on $[0,\infty)$. ...
0
votes
1answer
14 views

Determining density involving scaled beta distribution

Suppose $Y \sim \mathrm{Beta}(2,1)$. If $X = \theta{Y}$ (for some $\theta > 0$) how do I determine the joint density $f(x, \theta)$? Edit: the density for $Z$ is $2z$. Would it be correct to say, ...