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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51 views

Chance of having score of 63

Assume a batsman has an equal chance of getting a score of 1,2,3,4 and 6 and that he has 100% chance of eventually having a score greater than 63. What is the probability of the batsman having exactly ...
2
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1answer
67 views

Finding probability involving Poisson process.

Let $N(t)$ for $t\geq{0}$ be a Poisson process with intensity $\lambda > 0$. Now, let $X(t)$ be a process defined such that the arrival process of $X$ is every even numbered arrival of $N$. Then ...
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0answers
24 views

Geometric Distributions of Random Variables

Suppose that $X_1, X_2, ...$ are random variables that are independent and geometric, although perhaps with different parameters. Find as many of the following as is feasible: $X_1+X_2, X_1+X_2+X_3$, ...
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1answer
96 views

Secret Santa Probability Question

In a Secret Santa game, a group of N friends each write their name on a slip of paper and place these slips in a bowl. The slips are mixed and each person draws a slip from the bowl. If anyone draws ...
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0answers
26 views

Stationary Process Related To Balls in Urn

Suppose that you have an urn with 6 balls labelled '1' and 4 balls labeled '0'. At each time, you mix the balls and draw one, then record the value on the ball. You then replace it and add 2 more of ...
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1answer
26 views

Expectation of number of links to N selected nodes in a network

Take a directed graph denoted by its adjacency matrix $\mathbf{A}$. It is a probabilistic graph -- the nodes of $\mathbf{A}$ might be linked, and the entries are probabilities between 0 and 1. Say ...
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3answers
155 views

Making triangles out of a triangular lattice?

First off: Yes, I am well aware that this question has been posted before. However, the answer in that one was incorrect, so I decided to make another thread to gain more input, as well as provide ...
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1answer
26 views

Joint PDF of two independent normal distributions

I had this on my Probability final, and it stumped me. Exam is over and I still got a B, but here's the problem: Let X1, X2 be distributed as N(0,1) and N(0,9), respectively. Let Y1 = X1-X2, and let ...
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2answers
87 views

Playing card game probability problem!

So the problem is stated below: A card game of 52 cards are dealt between 4 people (including me), each receiving 13 cards. 4 of the cards are Aces; one is called Ace of Spades and the other Ace ...
0
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1answer
58 views

Mixed Distributions - Expectation and Variance

A bike has probability of breaking down $p$, on any given day. The repair cost of the bike, whenever it breaks down, is distributed as a Gamma random variable with shape $\alpha$ and rate ...
2
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1answer
38 views

Joint probability distribution of sum and product of two random variables

Let $X$ and $Y$ be two discrete random variables. I know the joint probability distribution of the vector $(X,Y)$, namely $P(X = x, Y = y)$ for all $x$ and $y$ in the sample spaces $\Omega_X$ and ...
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4answers
38 views

Law of total expectation

A man is trapped in a mine containing three doors. The first door leads to a tunnel that will take him to safety after walking for 3 hours. The second door leads to a tunnel that will return him to ...
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1answer
36 views

Central Limit Theorem approximation question

Suppose that the error, in grams, of a balance has the density $$f(x)=\frac{1}{4}e^\frac{-|x|}{2}$$ for $-∞<x<∞$, and that 100 items are weighed, independently of each other. Use the ...
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0answers
39 views

Product of two random variables - Resulting Distribution and Correlation?

Let $X \sim \mathcal{N}(0,1)$ and let $Z$ be a random variable independent of $X$ such that \begin{align*} P(Z=z) = \begin{cases}\frac{1}{2} & z=-1\\ \frac{1}{2} & z = 1\\ 0 & ...
1
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1answer
21 views

Simplifying Conditional Expectation with two Random Variables

In my introductory probability class I ran across these two expressions in a solution to a homework problem. X and Y are two random variables, and f(Y) is any function. ...
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1answer
48 views

Precise meaning/implications of “a random variable belongs to a space” almost surely.

As far as I understood, by saying a random variable/vector $X$ belongs to a space $S$ (or takes values in $S$), one means that the measurable function $X$ is $S$-valued: \begin{equation} ...
0
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1answer
58 views

How to define the sum of two random variables

I wonder how to define sum of two random variables properly. Suppose $(\Omega,\mathcal{F},P)$ is a probability space and $X,Y\colon \Omega\to \mathbb{R}$ are random variables. It is most certainly ...
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1answer
17 views

Probability of exactly one event occurring [duplicate]

Event A has probability of 0.7 and Event B has probability of 0.6. If A and B are independent, what is probability that exactly one occurs?
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1answer
40 views

expected value of harmonic mean

Let $X_1,...,X_n$ be i.i.d random variables with expected value $\mu$. Can we compute the expected value of the harmonic mean: $$ \frac{n}{\sum_{i=1}^{n}\frac{1}{X_i}} $$ thanks for helpers!
2
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1answer
31 views

Getting a full house with exactly 3 suits represented

How many ways can you make a full house with only 3 suits represented in the 5 card hand? My attempt: get the pair first: $$ {13 \choose 1}{4 \choose 2}$$ this allows us to pick any $2$ suits ...
1
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1answer
28 views

Let $X$ and $Y$ be discrete random variables with joint PMF supported on the points $(0, 0), (1, 1),(1, 0), (1, −1).$

Let $X$ and $Y$ be discrete random variables with joint PMF supported on the points $(0, 0), (1, 1),(1, 0), (1, −1).$ (A) Assign positive probabilities to these four points so that $X$ and $Y$ are ...
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2answers
43 views

Finding the pdf of $(X+Y)^2/(X^2+Y^2)$ where $X$ and $Y$ are independent and normal

$X$ and $Y$ are iid standard normal random variables. What is the pdf of $(X+Y)^2/(X^2+Y^2)$? I am guessing you would transform into polar coordinates and go from there, but I am getting lost. ...
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0answers
17 views

Timer action waiting probability

Say there was a 5 second timer. Every time the a second counter went over 5 seconds, something would happen if triggered beforehand. The command to do this something could come in at anytime between 0 ...
2
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0answers
37 views

Which of the following is always true for A and B

Given that: $ P(A) = 0.5$ $P(B) = 0.7$ $P(A \cap B) = 0.3$ I have to choose one option that is true... However they all seem to be false which means I am possibly making a mistake.. The only option ...
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2answers
43 views

Conditional Probability in Poker

I'm thinking of a ten person Texas hold'em game. Each person is dealt 2 cards at the start of the game. The question is: GIVEN that you have been dealt 2 hearts (Event B), what is the probability ...
0
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1answer
25 views

Characteristic function of an unkown sum of random variables

$X_1,X_2,...\sim Pois(7), $ and independent random variables. $Y \sim Geom(1/4)$ independent from the $X_i$. My question is the characteristic function of: $X_1+X_2+...+X_Y$ Can someone tell me ...
2
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1answer
41 views

$A$: set of Alice's frieds, $B$: Bob's friends, $C$: all people. Find $P(A \subseteq B)$ and $P(A \cup B = C)$

(Introduction to Probability, Blitzstein and Nwang, p.80) Alice, Bob, and 100 other people live in a small town. Let C be the set consisting of the 100 other people, let A be the set of people ...
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0answers
19 views

$(X,Y)$ is distributed uniformly over the square with vertices $(1,1)(1,-1)(-1,1)(-1,-1)$. Compute $\mathbb{P}(|X+Y|<1)$

$(X,Y)$ is distributed uniformly over the square with vertices $(1,1)(1,-1)(-1,1)(-1,-1)$. Compute $\mathbb{P}(|X+Y|<1)$ My ...
0
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1answer
41 views

Limit distribution of $X_n+Y_n$ if $X_n \overset{d}{\longrightarrow}X$, $Y_n \overset{d}{\longrightarrow}Y$ if $(X,Y) \sim N(\mu,\Sigma)$?

Let $X_n, Y_n$ be sequences of RV with $X_n \overset{d}{\longrightarrow} X$ and $Y_n \overset{d}{\longrightarrow} Y$ so that $\begin{pmatrix} X\\ Y \end{pmatrix} \sim N\left(\begin{pmatrix} ...
0
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1answer
25 views

Is it possible to obtain the following inequality?

Let $X,Y$ be two random variables such that $E\|X\|^{2n}\leq c_1^n,n\in\mathbb{N}$ and $E\|Y\|^{2n}\leq c_2^n,n\in\mathbb{N}$. Clearly, we have then $$E\|X+Y\|^2 \leq 2E\|X\|^2 + 2E\|Y\|^2 \leq ...
4
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1answer
101 views

Average time to fill boxes with balls

Let's have n users with each having a ball and m boxes. The users put their ball in a random box. It takes exactly 10 seconds for all balls to be put in a random box (independently to the number of ...
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2answers
57 views

Probability involving bread and jam!

SO, I drop a piece of bread and jam repeatedly. It lands either jam face-up or jam face-down and I know that jam side down probability is $P(Down)=p$ I continue to drop the bread until it falls jam ...
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0answers
17 views

Almost sure convergence of Chi-Squared variable

Setting $$X_1,X_2,\ldots \overset{d}{\sim} \mathcal{N}(0,1)$$ $$S = X_1^2 + \ldots + X_n^2$$ I would like to show $\frac{R_n}{\sqrt{n}} \rightarrow 1$ almost everywhere. I am using Borel-Cantelli ...
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2answers
42 views

Finer partitioning of the $[0,1]$ interval

Let $S\subset [0,1]$ . Lebesgue Density Theorem tells us that the density of $S$ is either $0$ or $1$ almost everywhere on the interval $[0,1]$. Define a $01$-transition to be a point $x\in [0,1]$ ...
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1answer
32 views

Convergence of sequence averages of rolls of a die to the expected value.

While I was reading about the expected value on the Wikipedia (http://en.wikipedia.org/wiki/Expected_value) one image attracted my attention: Legend of an image: "An illustration of the convergence ...
2
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1answer
77 views

Ultimate extinction certain if $\lambda \le 1$ in a (Poisson) branching process?

I have a branching process where each individual has a $$\sum_{k=0}^{\infty} \lambda^k \frac{e^{-\lambda}}{k!}$$ (Poisson) probability of producing k descendents. I want to show that the probability ...
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2answers
31 views

A question about different pairs that are formed from a set of 16 different balls such that…

I got the following problem: Given a set of 16 different balls, 8 are white and 8 are black. If we partition the set of balls into pairs of two different balls and let $X$ be a discrete random ...
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0answers
23 views

Find Cumulative and the probability density function of Y

Usually I would integrate the function $y=x^2$ from 2 to 1 and to find the probability density function but I need to show it in terms of t. How do I do this? Also is the cumulative distribution = ...
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0answers
44 views

Urn problem- distribution after all balls of x randomly selected colours are removed

Apologies for any notational problems or lack of clarity: I'm a linguist not a mathematician. Anyway, here goes: There is an urn with $n$ balls divided into $k$ colours, where the number of each ...
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2answers
37 views

Probability of drawing 2 white and 1 black ball

From a box containing 4 white and 6 black balls, 3 are transferred to another empty box. From new box a ball is drawn and it is black. What is the probability that out of 3 balls transferred 2 are ...
0
votes
1answer
65 views

Probability; Coin toss. Discrete Math.

I flip a fair coin, independently, 10 times, resulting in a sequence of heads (H) and tails (T). For each HT in this sequence, you win $3. Define the random variable X to be the amount of dollars that ...
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0answers
25 views

Sigma algebra which is not a topology [duplicate]

Can you please provide me with some $\sigma$-algebra which is not a topology?
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1answer
38 views

Mixed Conditioning - Two Normal Distributions

Let $Z \sim \mathcal{N}(0,1)$ and $Y|Z \sim \mathcal{N}(Z, 1)$. Show that $f_{Z|Y}(z|y)$ is a normal density, and find the parameters of this density. What I have so far: \begin{align*} ...
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0answers
24 views

Treatments of probability that gives the subject justice

I was watching a Joe Blitzstein lecture the other day on conditional expectation (https://www.youtube.com/watch?v=gjBvCiRt8QA). Mr Blitzstein noted a remarkable way of thinking about expectations, ...
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1answer
15 views

Discrete Math: Probability with subgraphs and coin flips.

Let $K_n$ be the complete graph on $n$ vertices, in which each pair of vertices is connected by an edge. For each each edge $e$ of $K_n$, we flip a fair and independent coin; if the coin comes up ...
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1answer
38 views

Show almost everywhere convergence for variable with Chi distribution

Setting $$X_1,X_2,\ldots \overset{d}{\sim} \mathcal{N}(0,1)$$ $$R_n = \sqrt{X_1^2 + \ldots + X_n^2}$$ I would like to show $\frac{R_n}{\sqrt{n}} \rightarrow 1$ almost everywhere. I have tried to set ...
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2answers
45 views

Discrete Mathematics Probability Review Question

Can someone please explain why the following question's answer is (a)? Assume that a newborn baby is a girl with probability p and a boy with probability 1 − p. Also assume that the genders of ...
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2answers
91 views

Probability that a bitstring of length $25$ will contain atleast two $1$s

We choose a bitstring of length $25$ uniformly at random. What is the probability that this string contains at least two $1$s? (a) $1 − \left(\frac12\right)^{25} − 25\left(\frac12\right)^{25}$ (b) ...
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1answer
31 views

$X,Y,Z$ are iid ~ $U(0,1)$, find $P(X>YZ)$ and $P(X<Y<Z)$

$X,Y,Z$ are iid ~ $U(0,1)$, find $P(X>YZ)$ and $P(X<Y<Z)$ I have no idea how to solve this problem, anyone could help me? Thanks
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0answers
19 views

Conditional expectations of joint normal distribution

$u_1$ and $u_2$ are jointly normal, with zero means, unit variances, covariance $\sigma _{12}$. I know $E(u_1|u_2)=\sigma _{12}u_2$, but why $E(u_1|u_2<c)= \sigma _{12}E(u_2|u_2<c)$ ?