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
9 views

Sampling from Bivariate Normal, Relation between two coordinates

Suppose the mean and covariance matrix of a bivariate normal distribution are respectively $(0,0)$ and $\begin{pmatrix}a^2 & c \\ c & b^2\end{pmatrix}$. Let $(x_1,~ x_2)$ be a datapoint ...
1
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0answers
8 views

Probability in selected group

35% of students are female. We know that if we select 20 from 200 students there are first 5 students who are male. What is a probability that 6th one is also male? First of all, I don't know if ...
0
votes
1answer
69 views

Definition of a random variable $\mathrm{Var}(X)$

So $\mathrm{Var}(X) = \mathrm{E}((X-\mu)^2)$, but how can you subtract a function $(X)$ by a value ($\mu)$? And does it make sense to square a function?
-2
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1answer
29 views

How do can i solve the integral, finding cdf [on hold]

Let $X$ be an exponential random variable with mean 1 and Y a uniform random variable between $0$ and $1$. Assume X and Y are independent and let $Z =e^{X/2}$ Find the joint cumulative ...
0
votes
1answer
21 views

Sum of random variables goes to infinity

I'm trying to show the following: Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d random variables with $\mathbb{E}[|X_1|]<\infty$ and $\mathbb{E}[X_1]=\mu$. Consider ...
0
votes
0answers
22 views

Probability with ordered statistics and exponential distribution involved

Assume that $X_1,X_2$ are independent random variables with exponential distribution with the same mean 100. Let $X_{(1)}=\min\{X_1,X_2\}$ and $X_{(2)}=\max\{X_1,X_2\}$. Calculate ...
3
votes
3answers
123 views
+50

Die that never rolls the same number consecutively

Suppose we have a "magic" die $[1-6]$ that never rolls the same number consecutively. That means you will never find the same number repeated in a row. Now let's suppose that we roll this die $1000$ ...
0
votes
3answers
28 views

joint density function of two independent random variables

Suppose that $𝑋_1$ and $𝑋_2$ are independent and follow a uniform distribution over $[0, 1]$. Let $π‘Œ_1 = 𝑋_1 + 𝑋_2$, and $π‘Œ_2 = 𝑋_2 βˆ’ 𝑋_1$. a) Find the joint pdf $𝑓_{π‘Œ_1,π‘Œ_2} (𝑦_1, 𝑦_2)$ ...
0
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0answers
17 views

Integal involving Normal and Log-normal PDF/CDF

I am following this example where the idea is to compute the failure probability using different approaches. The problem is given at the beginning. We have two random variables $R$ and $F$, following ...
1
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0answers
16 views

issue on conditional-expectation with crossed filtration

Why we have this equality ? $$\mathbb{E}[\ \mathbb{\hat{E}}(X(.)|\mathcal{F}_t)_G K(G) |\mathcal{F}_t] = \int_{\mathbb{R}}\mathbb{\hat{E}}(X(.)|\hat{\mathcal{F}}_t)_u K(u) dP_t^G(u)$$ For all ...
3
votes
2answers
24 views

An average of three calls arrive every $5$ min. Find the probability that exactly four calls will arrive during a $5$ minute interval.

An average of three calls arrive every $5$ min. Assuming a Poisson arrival rate, compute the probabilities of the following events: (a) exactly four calls will arrive during a $5$ minute interval. ...
0
votes
1answer
17 views

Find the conditional pmf of $Y$ given $X = 0$

Let $X$ and $Y$ have the joint pmf defined by $f(0, 0) = f(1, 2) = 0.3$, $f(0, 1) = f(1, 1) =0.2$ $(a)$ Tabulate the conditional pmf of $Y$ given $X=0$ $(b)$ Tabulate the conditional pmf of $X$ ...
0
votes
2answers
46 views

Number of outcomes with 3 distinct numbers rolling 4 dice.

Suppose you roll 4 distinct dice. I am trying to find: a) The number of outcomes with 3 distinct numbers b) The number of outcomes with 2 distinct numbers I just want to check that my reasoning is ...
0
votes
1answer
30 views

Mean value for a simple random variable

From a box with numbers from 1 to 90, 6 numbers are extracted without reintroduction. To play this "game", you have to pay 1 and you win 15 millions if you predict the 6 numbers (nothing in all the ...
1
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0answers
28 views

Specific Radon-Nikodym Derivative Interpretation

Suppose $(\Omega, \mathcal{F}, P)$ and $(\Omega, \mathcal{F}, Q)$ are two probability spaces. The Radon-Nikodym theory says that if $P$ is absolutely continuous with respect to $Q$, then there exists ...
0
votes
1answer
26 views

Calculating the mean and variance of continuous distribution

The main question was "A machine produces 2mm to 12mm usb sticks. Any usb greater than 10mm in size will need to be thrown away." Part A) Calculate the portion that needs to be thrown away, and I got ...
1
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2answers
36 views

Probability Proof about A and B

I have to formally prove that: $$P(A) = P(A\wedge \neg B) + P(A\wedge B)$$ so I did like this: $$P(A\wedge \neg B) + P(A\wedge B)$$ $$=P(A\wedge \neg B) + P(A)\cdot P(B)$$ $$=P(A)\cdot P(\neg B) + ...
3
votes
1answer
54 views
+50

Integrating a probability density function that only depends on the norm

I have a probability density function $f$ on $\mathbb{R}^3$ which only depends on the norm of a vector (that is, it takes the same value for $x,y$ if their length is equal). Let me call a region of ...
1
vote
1answer
1k views

Proof for Standard Deviation Formula for a Binomial Distribution

I understand the concept of standard deviation as the square root of the square of the mean of each sample value - the mean of the sample values. Here is the mathematical representation (I've solved ...
1
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2answers
14 views

Method for separating 'randomness' and 'non-randomness'

Let's assume I have a random two signals: Sin(t) R(t) Sin(t) is of course the trignometric function, but R(t) is a random process. So let's now assume I ...
0
votes
2answers
44 views

Can the sum be simplified? $\sum_\limits{x=y}^{\infty} {x \choose y}\left(\frac{1}{3}\right)^{x+1}$

Let: $$f(y) = \sum_{x=y}^{\infty} {x \choose y} \left(\frac{1}{3}\right)^{x+1}$$ Can this be simplified somehow? Is it a standard probability distribution? I can only get as far as: $$f(y) = ...
0
votes
0answers
18 views

Inequality for integral of complex valued functions

assume that $f$ is a complex-valued function acting on some probability space $(X,m)$ and $g$ a non-negative function defined on a same space such that $$ \lvert \int_A f \, dm \rvert \le \int_A g \, ...
1
vote
2answers
28 views

Optimization with a Probability

Imagine two points in $ℝ^2$ at $(-1, 0)$ and $(1, 0)$. You would like to walk from one point to the next in the shortest distance possible. However, there is a line segment coming from the origin to a ...
2
votes
1answer
745 views

Probability of three events occurring given correlation?

I am facing a problem that I cannot find the answer to. I have three variables, A, B and C. There are only two possibilities for each of these, A either happens or it does not, B happens or it does ...
0
votes
0answers
20 views

$N = Poisson(\lambda)$, $\{U_i\}$ iid $\implies (N_1, N_2) = Po(\lambda p_1)$x $Po(\lambda p_2)$

Let $\{N\}\cup\{U_i\}$ be independent random variables. $N = $ Poisson$(\lambda)$ $\{U_i\}$ iid, taking values in $\{1,2\}$, $\mathbb{P}[U_i = 1] = p_1$ and $\mathbb{P}[U_i = 2] = p_2$, $p_1 + p_2 ...
2
votes
0answers
29 views

Let $X$ and $Y$ be iid real-valued random variables. Show $P[|X-Y| \le 2] \le 3P[|X-Y| \le 1]$. [duplicate]

Found this question in The Probabilistic Method and tried for hours to prove it, but I'm not getting anywhere. Can anyone walk me through it? I see that if we can show $P[1 \le X - Y \le 2] \le P[|X ...
4
votes
2answers
22 views

lottery to pick a group while respecting pairs

I am running an event that will be oversubscribed, so I'd like to use a lottery to randomly pick the participants that will be accepted. (For example, 29 people want to attend, but I can accommodate ...
1
vote
1answer
33 views

Probability of chosen urns being filled after randomly throwed 2 balls k times

We have n urns. Repeat next process k times: choose 2 distinct urns, throw ball into each. What is the probability of choosing 2 urns with at least 1 ball in it? (e.g. we have 8 urns. Then choose 3 ...
0
votes
1answer
20 views

Cumulative distribution function (CDF) strictly less than

Suppose a distribution function for the random variable $X$ is given by $$F(x)=\left\{ \begin{array}{11} \hfill 0 \hfill & x \lt 0\\ \hfill \dfrac{x}{2} \hfill & 0 \leq x \lt 1\\ \hfill ...
0
votes
2answers
15 views

Clarification on variance and expectation of Y value

Let $X$ be a random variable where P(X = 1) = $\frac13$ P(X = 2) = $\frac13$ P(X = 3) = $\frac13$ Let $Y$ = (X βˆ’ 1)(X βˆ’ 2) be another random variable that depends on $X$. What is $E(Y)$? I ...
0
votes
3answers
34 views

If it has an emergency locator, what is the probability that it will be discovered?

Okay so here's the question Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60% ...
0
votes
1answer
17 views

Limit notation where variable does not approach anything

I was reading an example in my probability textbook that states a limit as $$\lim_{n}{P\left\{X \leq 3 - \frac{1}{n} \right\}}$$ where the RV $X=k$ is defined for $ k \in \mathbb{R}$ What exactly ...
7
votes
0answers
253 views

What is the Probability of Transmission Between Two Nodes in a Neural Network?

I have a network which is an ErdΕ‘s–RΓ©nyi graph. It is a simple neural network with degree 0.7N where N is the number of nodes. Each weight between neurons is 1/N, meaning that if node n has fired ...
-1
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0answers
13 views

Question about notation of brownian motion [on hold]

If $(B)_t$ is a sequence of Brownian Motion what does $B(t-1, t)$ stand for?
2
votes
4answers
54 views

Probability of $5\%$

Okay, I am playing a game. I have $5 \%$ chance of upgrading a weapon, so $95\%$ chance of it failing when I try to upgrade. How many times do I need to try to upgrade in order to guarantee a $100\%$ ...
1
vote
1answer
51 views

How to Express the Probabilities Associated with a Third Variable in a Hidden Markov Model?

Suppose I have an observation $Y_t$ that is conditionally dependent on $X_t$. (More specifically, Y is a series of observations emitted by an underlying hidden Markov state sequence X.) I can ...
1
vote
4answers
415 views

Figuring out probabilities with Hidden Markov Models

I'm really new to Math so sorry in advance if this question does not make sense. Also I cross posted this on stats.stackexchange.com also. Background: I'm trying to learn about hidden Markov models ...
2
votes
1answer
80 views

How do I do the derivative of characteristic function.

I have a uniform random variable $X$ in $[-b, b]$, and I got its characteristic function: $$\Phi_X(\omega)=E[e^{j\omega X}]= \int_{-b}^{b}e^{j\omega x}\cdot f_X(x)dx=\int_{-b}^{b}e^{j\omega x}\cdot ...
-1
votes
1answer
39 views

Distribution of sample maximum

I have $n$ balls. I choose $n$ balls uniformly at random with repetitions. Let's call $N(i), i \in \{1, \dots, n\}$ the number of times ball $i$ has been chosen. I would like to know how $max(N(i))$ ...
3
votes
1answer
22 views

Is a subsequence of an exchangeable sequence exchangeable?

Consider a finite sequence of random variables $X_1,...,X_n$ (1) SUFF COND: Suppose $X_1,...,X_n$ are exchangeable, meaning that the joint probability distribution of $X_1,...,X_n$ is equivalent to ...
1
vote
1answer
25 views

Find $g(x|y=\frac{1}{2})$, the conditional pdf of $X$ given $Y = \frac{1}{2}$ (Need confirmation)

Let X and Y be continuous random variables having the joint pdf $$f(x,y) = 8xy , 0\leq{y}\leq{x}\leq{1}$$ I found that the marginal pdf of Y is $f_2(y) = 4y - 4y^3$. Does $g(x|y=\frac{1}{2}) = ...
2
votes
3answers
55 views

Expected value of $g(X)$.

If $\mathrm{E}(X) = \sum_{x\in I} x\,\mathrm{P}(X=x)$, how can I deduce that $E(g(X)) = \sum_{x\in ?} g(x)\,\mathrm{P}(X=x)$? I don't see why it isn't $E(g(X)) = \sum_{g(x)\in ?} ...
0
votes
1answer
24 views

Probability using Brownian Motion

Assume that $B(t)$ is a Brownian motion and that $S(t)$ is defined as $S(t)=A\cdot e^{B(t)}$ for some positive constant $A$. Calculate the probability of the event ${S(3)>2S(1)}$. How could I go ...
0
votes
1answer
64 views

How to calculate P(A∩B), P(A∩C), P(B∩C) and P(AβˆͺC), P(BβˆͺC)?

A particular computer program outputs a number in {0,2,3} with probabilities as follows: $0 $ has a probability of $\frac{1}{3}$ $2$ has a probability of $\frac{1}{2}$ $3$ has a probability of ...
1
vote
0answers
11 views

Inequality regarding double integral of log supermodular function

I'm trying to prove that $\int\int\pi(v,w)f_{1}(w)f_{1}(v)dwdv\int\int\pi(v,w)f_{2}(w)f_{2}(v)dwdv>(\int\int\pi(v,w)f_{1}(w)f_{2}(v)dwdv)^2$ if $\pi(v,w)$ is log-supermodular and $f_1$ and $f_2$ ...
2
votes
1answer
44 views

Roll 6 dice, find the number of outcomes with 3 distinct numbers.

Suppose you roll six dice, how many outcomes are there with 3 distinct numbers. My attempt: First there are ${6 \choose 3}$ ways to choose these 3 distinct numbers. We consider 3 cases; Case 1: 3 ...
2
votes
2answers
1k views

Probability, conditional on a zero probability event

Is there a way to resolve probability of an event, given another event that never happens? Mathematically speaking the problem is: Given that $P(B) = 0$, $$P(A|B)=\frac{P(A \cap B)}{P(B)} = ...
8
votes
5answers
11k views

How many rolls do I need to determine if my dice are fair?

Roughly how many times do I need to roll a 6-sided die to feel confident that it's giving "fair" results? What about a 10-sided or 20-sided die? Note that I will be actually manually rolling physical ...
0
votes
0answers
31 views

Uniform probability bound involving two binomial random variables

Fix $c>1$. Does there exist number $m$ and function $f(\epsilon)$ such that for every $0<\epsilon<1$, $0<p<\epsilon$, and $n > f(\epsilon)$, we get ...
1
vote
1answer
23 views

Subtle Sample Space

I have been self-studying probability and statistics using Sheldon Ross's "A First Course in Probability" for a while, yet I still have problems on recognizing sample spaces in some probability ...