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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-4
votes
1answer
34 views

Given any 40 people, at least four of them were born in the same month of the year

Given any 40 people, at least four of them were born in the same month of the year. Why is this true?
0
votes
1answer
19 views

Calculate the characteristic function $\varphi_W$ of W

$p(x)=xe^{-x}$ for $x\geq 0$ or $0$ otherwise. I tried to substitute $e^{-x}$ but then i found there is still a $x$ in front.
3
votes
1answer
24 views

Let $E(X)=\mu$ and $\operatorname{Var}(X)=\sigma^2$. If $E(Y|X)=a+bX$, find $E(XY)$ as a function of $\mu$ and $\sigma$.

I can't figure out the answer for a question on my econometrics course. Somehow it seems simple, but still I can't seem to figure it out. Maybe I am thinking the wrong way about it. Could someone ...
0
votes
0answers
8 views

Conditional entropy and independent conditioning variables

Let $X,Y,Z,Y',Z'$ be random variables where $(Y,Z)\sim (Y',Z')$, $Y$ and $Z$ are independent, while $Y'$ and $Z'$ are not. Is $H(X|Y,Z)=H(X|Y',Z')$? It seems whether $p(Y,Z)$ factorises or not does ...
1
vote
1answer
28 views

Find a continuous PDF on $[0,6]$ for given probabilities

Find a continuous probability density function $f$ on $[0,6]$, such that $\mathbb{P}([0,2]) = 0.6$, $\mathbb{P}([1,4]) = 0.5$ and $\mathbb{P}([3,5]) = 0.2$. After some calculations I came up with ...
1
vote
0answers
31 views

If $P(X_1 < X_2)$, what is $P(X_1 < X_2 \cap X_1 < X_3)$?

Say $X_i$ can have a real value in the range [1,100]. All $X_i$ are independent of each other and all values are equally likely. So then $\mathbb{P}(X_1 < X_2) = \frac{1}{2}$, right? But then, ...
1
vote
0answers
16 views

Poisson Process with continuous rate, Finding Conditional Number of Arrivals

Poisson with customer arrival to the shop rate given by $\lambda (t)=16-(t-4)^2$ Calculate $P(N(5)-N(3)=40|N(4)=70)$ where $N(i)$ means the number of arrivals in the first $i$ hours. The shop ...
0
votes
1answer
11 views

Probability of a vector of normal distribution

Given the set of vectors $\{\mathbf{g}^{1}, \ldots, \mathbf{g}^{N-1} \}$ where $\mathbf{g}^{i} \in \mathbf{R}^M$. Assume that $N \leq M$ and elements of $\mathbf{g}^{i}$ follows normal distribution, ...
1
vote
3answers
44 views

Find the distribution of $Y = -\log (1-X)$ given that $X\sim U(0,1)$.

If $X \sim U (0,1)$ then if we define a new random variable $Y=-\log (1-X)$ then what will be distribution of $Y$. Please explain.
0
votes
0answers
21 views

Appropriate distribution for set of probabilities $p_1 ,…, p_n$

I am doing some evaluation of a system, that has set of probabilities $p_i$ $i= \in \{1,...,N\}$, I need to model them as random variables such that : $$ \sum_i p_i \leq 1$$ and $$ 0 \leq p_i \leq 1 ...
2
votes
0answers
14 views

Showing a relation between binomial and negative binomial analytically

If $X$ is binomial random variable $B(n,p)$ and Y is negative binomial $(r,p)$, How can I show that $F_X(r-1) = 1- F_Y(n-r)$. While it is possible to show that using the definition of binomial and ...
1
vote
1answer
16 views

Derive the distribution of $Z$ given two identically and independently exponentially r.v.s?

$$Z=\frac{X}{X+Y}$$ $(X,Y)$ are iid r.v.s with $$f(x)=\lambda e^{-\lambda x}$$ We are asked to condition on $Y$ to derive the distribution of $Z$; $F(t)$ and $f_Z(z)$. I don't know where to ...
2
votes
1answer
22 views

What is $\operatorname{Pr}\{X_j=0|X_i=k\}$ [on hold]

Suppose $u_n=\operatorname{Pr}\{X_n=0|X_0=1\}$ What is $\operatorname{Pr}\{X_j=0|X_i=k\}$, where $\{X_n\}$ is a branching process and $k\geq 0$, if we were to write the answer in terms of the ...
3
votes
0answers
23 views

Markov Chain: Steady State Distribution.

A total of $M$ balls are divided between two urns A and B. A ball is chosen uniformly at random. If it is chosen from urn A then it is placed in urn B with probability $b$ and otherwise it is returned ...
0
votes
1answer
10 views

If $P$ is a transition matrix, and $m_{ij}$ the mean return time, how to show $m_{ij} = 1+ \sum_{k \neq j}P_{ik}m_{kj}$?

If $P$ is a transition probability matrix of a finite state regular Markov Chain, and $m_{ij}$ is the mean return time, how can I show that $m_{ij} = 1+ \sum_{k \neq j}P_{ik}m_{kj}$? It seems rather ...
-1
votes
0answers
21 views

show that $Y_1$ is unbiased for $\theta$ and find its variance

Let $X_1,\ldots,X_n \stackrel {\text{iid}} {\sim} \text{exponential}(θ)$ $$Y_1= \frac {X_1+3X_2+5X_5} {9} $$ $$ Y_2= \sum_{i} X_i$$ Show that $Y_1$ is unbiased for $\theta$ and find its variance. ...
1
vote
1answer
22 views

Expected value problem: flip $6$ fair coins before we obtain $3$ heads and $3$ tails?

How many times on average (expected value) must we flip $6$ fair coins before we obtain $3$ heads and $3$ tails? I know I need $∑ xp(x)$. I just don't know how to apply it.
0
votes
1answer
22 views

Why does $E(C\cdot \epsilon\; \vert\; C\cdot X) = E(C\cdot \epsilon\; \vert\; X)$?

Let $C$ be an $n\times n$ matrix, $X$ is $n \times k$, $\epsilon$ is $n \times 1$ This is taken from a simply proof of strict exogeneity in an Econometrics textbook by Hayashi. The explanation he ...
0
votes
0answers
7 views

When does a periodic but positive recurrent markov chain have a limiting distribution

So I know it's a fact that an aperiodic, finite state, irreducible (so positive recurrent) markov chain has a unique stationary distribution which is limiting. However, I am curious if there is a ...
1
vote
2answers
33 views

What is the probability that a psychic correctly “predicts” the outcome of at least 5 out of 10 coin flips?

Assume the psychic is actually just randomly guessing on each flip. The attempt: let E be the event in question number of outcomes per flip = 2 chance of correctly guessing the correct outcome = ...
0
votes
1answer
23 views

Troubles With The Beginning

The following is the question I'm having a bit of troubles starting: Musicnotes.com sells sheet music in the following genres: rock jazz, new age, and country. An experiment consists of recording the ...
1
vote
0answers
19 views

How to Calculate Covariance of Branching Process

Suppose we have a branching process $\{X_n\}$, where $X_0=1$. How would I go about calculating the covariance $\operatorname{Cov}(X_j,X_i)$ for $i\leq j$? Not sure how to start, so hint would nice to ...
0
votes
1answer
18 views

Prove a conditional distribution is uniformly distributed across a given interval?

$X$ and $Y$ are independent random variables identically exponentially distributed with $\lambda$. Take $Z=X+Y$. Show that $(X|Z=z)$ is uniformly distributed over $(0<x<z)$. Then, find ...
0
votes
0answers
18 views

How to determine the limiting distribution of a Markov Chain which can only increment up or down a state at every stage?

I have a random walk Markov chain that has states from $0$ to $N$. The conditions are that when the chain is at $0$, the chain will go to state $1$ with probability $1$. When the chain is at state ...
0
votes
0answers
57 views

How many ways are there to choose one-half dozen donuts from $9$ varieties so that there are exactly $4$ glazed? [on hold]

How many ways are there to choose one-half dozen donuts from $9$ varieties so that there are exactly $4$ glazed? How should I approach this problem? Okay I think it's C(10, 2) because I already have ...
2
votes
1answer
30 views

Finding expression for probability given its PGF

Consider the probability generating function for a random variable $X$: $\varphi_X(s)=\dfrac{7-3s}{15-14s+3s^2}$ Find an expression for $P(X=k)$, for $k\in\mathbb{N}$ My attempt was to break ...
1
vote
1answer
12 views

Doubt with Notation on Conditional Expected Value Demonstration

I´m having trouble writing a demonstration for the Conditional Expected Value using $\sigma$-algebra. I know its really simple and actually logic but I just can´t find the way to write it. Hope anyone ...
1
vote
2answers
27 views

Probability of a fair sequence of tosses ending on two successive tails given the first toss was a head?

Suppose a coin is tossed repeatedly until either two successive heads appear or two successive tails appear. Then, assume that the first coin toss results in a head. I would like to find the ...
1
vote
0answers
30 views

Probability Random Variables Fall in an Interval

I've been trying to figure out a counting problem and can't wrap my head around how to calculate the probability. If we let $X_{1}, . . . , X_{10}$ be independent random variable with a uniform(0,1) ...
1
vote
0answers
23 views

Mean distance of random points on a rectangular grid

I have a $N\times N$ grid of side $L$. Each gridpoint can be black or white and a ratio $r$ of the points is black. I want to predict the mean distance between two black points. The most appropriate ...
0
votes
0answers
10 views

Failure boundary for simple routing problem

As an absolute beginner concerning probability theory I am currently trying to solve the following problem: Given a grid that has $x$ columns (here $x = 4$) and $y$ rows (here $y = 5$), we insert a ...
0
votes
0answers
15 views

Continuity of random variable as function of a random variable

Suppose, we are given a continuos random variable $X$ and a continuous and nondecreasing function $f$. Can it be shown that a second random variable $Y=f(X)$ is continuos on the support of $X$? What ...
4
votes
2answers
120 views

r distinct balls in N boxes

If r distinct balls are distributed at random into N (N ≤ r) boxes, what is the probability that box 1 will receive exactly j balls ( 0 ≤ 𝒋 ≤ r)? my solution is [sample space] =$ N^r $ ...
0
votes
5answers
33 views

Probabilty derivation using axioms

$$P((A \cap B^c) \cup (A^c \cap B))=P(A) + P(B) -2P(A \cap B).$$ I need to show this holds. I see it with Venn diagrams but I need to show it using only the axiom, for the union of two disjoint sets: ...
0
votes
1answer
26 views

Conditional Probability

An incoming freshman Mark believes that he has a 25% chance of earning a GPA in the 3.5 to 4 range, a 35% chance of graduating with a 3.0 to 3.5 GPA and 40% chance of finishing with a GPA less than ...
0
votes
0answers
20 views

Obtaining a percentage from a range [on hold]

In statistics, having an $[a,b]$ range how can I obtain the percentage of the distribution that is included in it? Thanks.
1
vote
1answer
8 views

Can I calculate the probality of a test being true $Pr(T)$ from $Pr(T|V)+ Pr(T|not V)$ if I know that Pr(V)+Pr(not V)=1?

If have $T$="The virus test is positive." and $V$="There really is a virus." and I know that $Pr(V)+Pr(\bar{V})=1$, can I then say that $Pr(T)=Pr(T|V)+Pr(T|\bar{V})$ and how do I show that ...
1
vote
2answers
34 views

How to compute a “luck percentile” from a set of random numbers or die rolls

I think it's easiest if I start with my actual use-case: In a video game (XCOM), soldiers shoot at aliens. When they do, they have a % chance to hit. Hitting deals damage. I want to look at each ...
1
vote
2answers
13 views

Survival bias and probability

Imagine the following situation: A new virus is discovered that is believed to have infected 20% of the population. Anyone infected with the virus has a chance of 50% of dying in their sleep every ...
-1
votes
0answers
16 views

renewal process and probability(exercice) [on hold]

Let $N_t$ a renewal process. Let $A_t=t-S_{N_t-1}$, $S_{N_t}=X_1+...+X_{N_t}$ with $X_i$ the jumps moments. Let $Z_A(t)=P(A_t \leq u)$ 1) How to show $P(A_t \leq u |X_1=x)=P(A_t \leq u |X_1 \geq t)$ ...
0
votes
1answer
25 views

Marginal distribution from a Poisson distribution where intensity is exponentially distributed?

Given that $N$ is Poisson distributed with a random intensity $Y$, the conditional distribution of $(X|Y)$ is defined as, for $n=0,1,\dots$ $$P[N=n|Y=\lambda]=e^{-\lambda}\lambda^n\frac{1}{n!}$$ $Y$ ...
0
votes
2answers
20 views

Probability same outcome 3 times in a row.

I am doing some old exam questions - and I don't know the answer, can some one calculate the result and show how you did it?
0
votes
1answer
24 views

How do you picture: $\Pr(B|A)$ shrunk down by $\Pr(A)$?

I do not understand how to picture and visualise the following explanation: $\color{green}{[P1.]}$ Suppose you were to grab the edges of $A$ and stretch it out so it covers all of $\Omega$. $B$ ...
4
votes
1answer
21 views

Ehrenfest Chain: stationary distribution

In the Ehrenfest Chain model: There are M balls which are divided between urn A and urn B. At each stage, if a ball is chosen, then it would be moved into a different urn. Let $X_n$ be the # of ...
2
votes
1answer
22 views

Probability matrices in an online game or how to approach matching players to maps to achieve better user experience

Probably I had nonstandart question, but I hope to find some help and valueable advice. Assume I have an online game with $n$ players (let's say $n$ is about 100.000). There's also $m$ maps ($m$ is ...
0
votes
0answers
14 views

Probabilities of each waitlist person

Coming from this, $10$ Applicants for a exclusive club membership. I found that you can use total probability to consider existing and old members leaving/returning as members in the club, ended up ...
0
votes
1answer
16 views

Expected Number of Draws without Replacement

An urn contains a white balls, b blue balls and c red balls. Balls are drawn one by one without replacement. Find the expected number of draws needed to get all 3 colors.
1
vote
1answer
22 views

Difference between $F_X(x)$ and $F(x)$ in probability?

What is this difference in notation between $F_X(x)$ and $F(x)$? (where $F(x) = P\{X \leq x\}$ Thanks.
0
votes
0answers
10 views

Expected values of Hermite polynomials under Gaussian distribution

On Wikipedia there's a nice result stating that $$E[He_n(X)]=\mu^n,$$ where $He_n$ is the (probabilists') Hermite polynomial of order $n$ and $X$ is a $N(\mu, 1)$ random variable. I'm interested in ...
0
votes
2answers
20 views

Find PDF on $[0,6]$ such that $P([1,3]) = 0.5$

Find a probability density function $f$ on $[0,6] \subset \mathbb{R}$, such that $\mathbb{P}([1,3]) = 0.5$ That is we need to find an $f$, such that $\int_{[0,6]} f(x)dx = 1$ and $\int_{1}^{3} ...