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
24 views

calculating X, Y, Z random variables

Suppose X, Y, and Z are random variables that each take the value 0 or 1. If P(X=0,Y=1,Z=0)=1/3 and P(X=0,Y=1,Z=1)=1/4, what is the value of P(X=0,Y=1)? I am trying to calculate this but I am really ...
2
votes
1answer
22 views

arbitrage free price in martingale measures

Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ...
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votes
1answer
23 views

Understanding $Po(np)\{A\}$ probability notation

I am trying to read a textbook on probability and am already stuck on what must be basic notation. It says Thus, for example, if $A$ is any subset of $\mathbb{Z}^+$, it follows that for some ...
1
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1answer
29 views

Odds of winning more than 50% of many bet of different %

I made a bet with a friend and I would like to know if I'm ahead or not. We have a package of 6 games. Each game have a different probability of a team to win. If it's a tie 3-3, it's a push. If it's ...
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1answer
24 views

How are the waiting times distributed, poisson process.

I am wondering how the waiting times are distributed for the poisson process, conditioned on a number of events by time t. Look at this theorem: Here, the S's are the sum of the waiting time to ...
1
vote
1answer
21 views

Ping Pong Winning Probability (World Series)

You are playing ping pong with a friend and your chance to win any point is P. This is a world series. Find the probability that you score 4 points before your friend has a score of 4. Evaluate this ...
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1answer
26 views

probability for two vectors to lie on different regions created by hypeplane

Suppose we have two vectors $v_i,v_j$ and there is one hyperplane whose normal is chosen uniformly from the unit sphere. Then what will be the probability that $v_i$ lies on one side and $v_j$ lies on ...
3
votes
1answer
23 views

Probability of swapping elements in an array

Given an array with n numbers (n>30). Perform the following steps: Compare the first element with the second element. If the first one is greater than the second one, swap their positions in the ...
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2answers
24 views

Covariance of dependent random variables from a Poisson process

Question: Given a Poisson process $N(t),t≥0$ with rate $λ$, calculate the covariance of $N(2)$ and $N(3)$. Attempt: So clearly $N(2) \sim Po(2\lambda)$ and $N(3) \sim Po(3\lambda)$. So, ...
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0answers
13 views

Probability in a lottery game

In this lottery you guess 7 out of 33 numbers plus 1 out of 6. What is the chance of guessing 7+1 correctly?
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0answers
14 views

Difference between conditional and intersection in probability.

I am having hard time figuring out if it is a conditional probability or an "and" probability under the following types of problems. When a student is absent, the probability of the student being ...
0
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3answers
50 views

Solve my probability doubt? [closed]

A parent gives birth to two children. One of the child is surely a male, what is the probability of having both male child? Common answer 1/2 Actual answer 1/3
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0answers
23 views

Frequency function binary tree with 4 nodes

In the experiment of choosing a rooted binary tree with 4 nodes. Let X be the number of leaves and Z, the height of the tree (longest path [# of edges] from the root to a leaf). Use equally likely ...
0
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0answers
28 views

Probability: Disease and Diagnosis

The probability of occurrence of a certain disease in a population is $1/101$. A diagnostic test has $9$ out of $10$ chances to detect the disease when the tested subject is actually affected. On the ...
0
votes
1answer
21 views

Expected value minimum of descrete and continuous uniform distribution

I need the expected value of a minimum, normally this is not a problem at all but in this case it is the minimum of a uniform discrete and a uniform continuous distribution. Let $X$ be a random ...
1
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2answers
26 views

How do I calculate variance for sum of dice?

I'll post my work, but I'm not sure how to calculate variance. The question asks for the expected sum of 3 dice rolls and the variance. I think I got the expected sum. Any help would be awesome :) ...
0
votes
1answer
47 views

Fair Die tossed twice, random variables

A fair die is tossed twice. Let $d_1=\text{value of die on roll 1}$ and $d_2=\text{value of die on roll 2}$ Let $X=d_1+d_2$, the sum of the faces; $Y=\max\left\{d_1,d_2\right\}$, the maximum of the ...
2
votes
3answers
38 views

Marginal density function of a new variable

Question: Let $X \sim U[0,1]$ and $Y \sim U[0,1]$ be independent random variables. By considering the random variables $U=Y$, $V=XY^{2}$, or otherwise, find the probability density function of $V$. ...
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0answers
19 views

Conditional Probability Question - on route availability

Hey Guys I am seemingly stumped with this question I have gotten involving conditional probability and routes Suppose route $A$ to $B$ is available 0.5 of the time An alternative route to B from A ...
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0answers
53 views

Donsker for randomly stopped processes

A question regarding Donsker's invariance principle. Donsker states that if $X_1, X_2, ...$ are independent and identically distributed with mean $0$ and variance $\sigma^2$ and if $S_t^n$ is the ...
2
votes
3answers
160 views
+100

Prove that there are two frogs in one square.

A certain chessboard is infinite in size. There is a frog sitting in the center of every square. After a certain time, all the frogs jump such that They may jump to any possible square in the ...
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1answer
27 views

Probability and Percentage

If there is a 70% chance a ball game will be played on any given day what is the probability of a ball game being played Monday through Friday?
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0answers
33 views

If X is a Poisson variate and $ p(X = 3) > p(X = 2)$ [closed]

$X$ is a Poisson variate and $ p(X = 3) > p(X = 2)$ Then how to find the the minimum value of the mean.
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0answers
31 views

Norm of a projection in $L_p$

Let $\mu$ be a probability measure on $[0,1]$ and let us define the projection $P$ on $L^p(\mu)$: $$P \; \colon f \mapsto \mathbb{E}(f){\bf 1}$$ (where ${\bf 1}$ is the constant 1 function). What is ...
0
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0answers
22 views

Conditional Probability and conditional expectations [closed]

Let $X_i$, for $i\geq 0$, be independent and identically distributed random variables with p.m.f. $p(j)=\Pr(X_i=j), j=1,\cdots,m$, where $\sum\limits_1^m p(j)=1$. Find $\text{E}[N]$ where ...
1
vote
1answer
43 views

Proving that Markov Chain Monte Carlo converges

I am trying to understand how the very basic Markov Chain Monte Carlo approach works: We try to approximately calculate the expected value $E_{\pi(x)}[X]$ by drawing sequential samples from a Markov ...
0
votes
1answer
40 views

Probability: put 20 distinct balls randomly in 12 urns

You put 20 distinct balls randomly into 12 urns. What is the probability of having 3 urns with 4 balls each and 4 urns with 2 balls each (the other 5 urns are left empty). For my sample space I have: ...
-1
votes
0answers
17 views

Finding P(L=1, A= 0) and P(L=1|A=0) [closed]

P(A = 0) = 1/4 P(B=1 | A=1) = 4/5 P(B=1 | A=0) = 2/5 P(L=0| B=0) = 1/3 P(L=1 | B=1) = 1/4 How would I find P(L=1, A = 0) and P(L=1|A=0)? I have been thinking about this for a while and I can't ...
0
votes
1answer
23 views

Finding P(B=1) using previous given information

So I have this given information: P(A = 0) = 1/4 P(B=1 | A=1) = 4/5 P(B=1 | A=0) = 2/5 P(L=0| B=0) = 1/3 P(L=1 | B=1) = 1/4 So my thought process is this: Bays Thm is this P(A|B) = ...
-1
votes
1answer
13 views

P(B=0|A=1) = 1 - P(B=1|A=1)? [closed]

I got a question. If I know P(B=1|A=1) = 4/5, and B and A can either be 1 or 0, is the probability of P(B=0|A=1) = 1 - P(B=1|A=1)? Any tips would be great thanks!
1
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1answer
25 views

Bayes Theorem, The Law of Total Probability, and Trees that Don't Grow.

" A doctor is concerned about the relationship between blood pressure and irregular heartbeats. Among her patients, she classifies blood pressures as high, normal, or low and heartbeats as regular or ...
0
votes
1answer
40 views

Inferring symmetry of a distribution from its marginals

Let $X=[X_1,\ldots,X_n]$ be a continuous random vector of size $n$ with density function $f_X(x_1,\ldots,x_n)$. If all the marginals \begin{align*} \int \ldots \int f_X(x_1,\ldots,x_n)\, ...
0
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0answers
24 views

Function of two Random Variables

Let X and Y be two independent random variables with the same probability density function given by: $f(x)= e^{-x}$ if $0 < x < \infty$, $f(x) = 0$ otherwise Show that $g$, the probability ...
1
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1answer
34 views

{Probability}: choosing keys from a pool without replacement

The OP is trying to understand the following question. The OP understand that if you can always write out the term $$P(X=k) \implies (1-\frac{1}{N})(1-\frac{1}{N-1})\cdots(1-\frac{1}{N-k+1}),$$ ...
1
vote
2answers
61 views

Probability Equation That I am missing here

1) What is the probability that out of giving birth to 6 kids, 3 of them will be boys? The answer is 50% chance? Don't ask me why but I just logically see that it is 50% so why? 2) But what about ...
0
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0answers
24 views

indepence transitive property?

For the events A and B are independent and B and C are independent is A and C independent I used coin tosses to try to model this with A = H B = T and C = H in seperate fair tosses I get that they ...
0
votes
1answer
36 views

probability of playing music player on shuffle and listening to every song.

I have a few problems I am trying to work out but I am not totally confident in my answers: The problem is such: Suppose you have a playlist consisting of four songs. You play your playlist in ...
-1
votes
1answer
33 views

$E[\hat{\theta}_{MME}] = E[\frac{1- 2\overline{y}}{\overline{y}-1}] = \int_0^1 \frac{1- 2\overline{y}}{\overline{y}-1}(\theta+1)y^\theta dy$..?

Let $Y_1, Y_2,\dots , Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
0
votes
1answer
31 views

Gaussian vectors and covariance matrix.

The following is a part of a question I was given in stochastic processes course. It goes like this - I am given a series of gaussian iid random variables $\{V_i\}_{i=1}^N$ , the variable $X_0 \sim ...
1
vote
1answer
18 views

basic conditional probability proof

I having trouble with the following proof: $$P((A \cap B) \mid B) = P(A\mid B).$$ I get that $P(A\mid B) = P(A \cap B) / P (B)$, but I am unsure of how to proceed.
2
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1answer
31 views

Markov Chain Coin Flips

If you flip a fair coin $n$ times, what is the probability that the $n$th flip is the first time in your series of flips that completes a pair of consecutive heads?
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2answers
41 views

Find the generator of Markov Process

Homework question: Consider the Markov process $X_t=B_t-t^2+t$ where $B_t$ is the Brownian motion. Find the generator $Q$ of this process. I am completely confused how to find the generator for ...
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0answers
11 views

Are there any models that have mean $\sqrt{t\log(t)}$?

R. Arratia (The Motion of a Tagged Particle in the Simple Symmetric Exclusion System on Z) shows in theorem 2 that for step initial condition in the SSEP, the position of the lead particle, $x_1(t)$ ...
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0answers
35 views
+50

Can't find gradient for MLE for mult-class logistic regression

$$P(k | x_i;w)= \frac{exp(w_k^tx_i)}{\sum_{j=1}^K exp(w_k^tx_i)}$$ $y_i^k$ is a vector that uses 1-of-k encoding. Thus, if $y_i=k$, then the vector $y_i$ has a 1 in the kth spot and a 0 everywhere ...
1
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1answer
31 views

Die Question with Random Variables

I have a homework question: Roll a fair die, and let d be contained in $\{1,2,3,4,5,6\}$ . Then sample $d$ independent uniform random variables on $[0,1]$ and let $Y$ be the maximal of these random ...
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0answers
29 views

Why is the expected number of steps to finish this game $O(n^2)$?

Say I play a game. I start at a number line at 0. At each step, I increment by 1 with probability 1/2 and decrement by 1 with probability 1/2. The game ends when I am at integer $n$. Why do I take an ...
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votes
1answer
52 views

Finding $E[X^2]$ if $E[X] = \frac{\pi k}{4}$ [duplicate]

We try to approximate $\pi$ by choosing random points in a square and seeing if they lie within the inscribed circle. The probability that a point is in the circle is $\frac{\pi}{4}$. Suppose we ...
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2answers
22 views

Conditioning on a joint exponential distribution

I'm working on a problem from Hogg (7.3.4) where there is a joint pdf. $$f(x,y) = \frac{2}{\theta^2}e^\frac{-(x+y)}{\theta}$$ Valid for $ 0 < x < y < \infty$ As part of the problem, I need ...
0
votes
1answer
15 views

Variance formula - need verifying [closed]

$$D^2(X)=D^2(E(X|Y))+E(D^2(X|Y))$$ Can someone verify that this is true and why?
1
vote
1answer
37 views

Probability and Combinations

In a family with 6 children, a. What is the probability of having three children of each sex? b. What is the probability of having four of one sex and two of the other sex? I know in this problem ...