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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2
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0answers
6 views

Variance of the random variable $|X \cup Y|$?

Let $X$ and $Y$ be random subsets of $\{1, 2, \dots, k-1, k\}$ picked uniformly at random from all $2^k$ subsets, independent of each other. What is the variance of the random variable $|X \cup Y|$?
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0answers
3 views

Equiprobable Model combinations

In a question in our statistics project, there is a set of balls, numbered 1 to 10, each ball is equally likely to be selected, making the sample space S = {{i, j, k, l} : 1 ≤ i, j, k, l ≤ 10 where ...
-1
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0answers
13 views

Conditional probability of having a flu shot, based on symptoms

During one flu season, a medical researcher encouraged all of her patients who were older than 50 to have a flu shot. She then kept records of all patients who displayed flu like symptoms over the ...
0
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0answers
6 views

Combinatorics/Probability Insurance accident

An insurance company classifies people as normal or accident prone. Suppose that the probability that a normal person has an accident in a specified year is 0.2 and that for an accident prone person ...
0
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0answers
11 views

Bayesian update and discontinuous cdf

I have an uniform random variable $x\sim U(0,1)$. I receive a signal $z$ about $x$ that is given by $$ z=y(x)+\varepsilon $$ where $\varepsilon\sim U(-\frac{1}{2},\frac{1}{2})$ (independent from $x$) ...
0
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1answer
14 views

What kind of information is Fisher information?

Suppose we have a random variable $X \sim f(x|\theta)$. If $\theta_0$ were the true parameter, the the likelihood function should be maximized and the derivative equal to zero. This is the basic ...
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0answers
9 views

expected and renewal process

Let $N_t$ a renewal process with $T_i$ the jumps. I know $U(t)=E(N_t)$ on [0,t]. Let $B_t=T_{N_t}-t$ and $G_t(u)=P(B_t \leq u)$ How to show on ]t,t+a]: $E(N_{t+a}-N_t)=\int_0^a U(a-u)G_t(du)=(G_t ...
0
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0answers
10 views

Solving an equilibrium equation for a markov chain

I have this statistical equilibrium equation at state $(j_1,j_2)$ from a 2-Dimensional markov chain given in the diagram:2-D Markov chain. I don't know how to solve this equation or how can i ...
0
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0answers
19 views

Average number of steps to return to the origin of a random walk on a 2-d lattice.

Suppose I have a random walker on a 2-d square lattice with periodic boundary conditions with equal probability of going in any of the four directions. The walk ends when the walker reaches the point ...
0
votes
1answer
29 views

From a group of 20 hunters, 5 hit a target with probability 4/5, 7 with probability 3/5 and 8 with probability 1/2. [on hold]

From a group of 20 hunters, 5 hit a target with probability 4/5, 7 with probability 3/5 and 8 with probability 1/2. A hunter taken at random shoots, without hitting the target. Which is the ...
2
votes
2answers
15 views

Basic Probability Question about number selected at random intervals with a modulus

This is my thought process on solving this question, but the answer sheet is not available and I am unsure about my thought process. Please correct me if I am wrong. Question: A number x is selected ...
0
votes
1answer
23 views

The probability to log on a computer from a remote terminal is 0.7.

The probability to log on a computer from a remote terminal is $0.7$. Let $X$ denote the number of attempts that must be made to gain access to the computer. Find: (a) The distribution of $X$ and ...
1
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0answers
16 views

Randomized Quick Sort and Partition Probability?

We know about Quick Sort and Randomized Version and Partition. I ran into a Fact when I read my notes. Let $0 < a < 0.5$ be some constant. We have an $n$-element array as input. Randomized ...
3
votes
3answers
25 views

Basic question. Does not involve permutations/combinations. $3$ mailboxes containing $3$ letters

My teacher explained this problem to us - "There are $3$ mailboxes. $3$ people put letters in at random. There is no preference for any of the $3$ mailboxes. Compute the probability that each mailbox ...
0
votes
1answer
17 views

calculating probability of an event

I have encountered a question The weather report says that there is a P probability of rainfalls today. Raj has to step out for a meeting at the office, and would like to know the probability ...
0
votes
1answer
21 views

central limit theorem for function of random variables

Let's say you have $X_1,...,X_n$ observations of a RV X, which is distributed according to some arbitrary prob. function. Further there is a deterministic function $f(X)=Z$, $f: X \rightarrow [-1,1]$ ...
3
votes
1answer
51 views

How to find a probability?

Two ships, independently arrive, at the port in any time within $24$ hours ($0-24$h). Every moment of arrival of ships is equally possible within $24$ hours. If the port can handle only one ship and ...
0
votes
2answers
34 views

Buffon needle problem , scenario $\ell>d$

suppose we have the classic problem of buffon's needle , let $\ell$ be the length of the needle and $d$ the distance between the parallel lines . I have solved the case which $\ell \leq d$ and i ...
0
votes
0answers
10 views

renewal process and renewal function

Let $N_t$ a renewal process and $T_i$ the jumps. $T_i=X_1+...+X_i$ and let F the distribution function. Let $(F \star F)(t)=\int_0^t F(t-u)F(du)$ and $U(t)=\sum_{n=0}^{\infty} F^{\star n}(t)$ the ...
2
votes
4answers
35 views

pdf is defined as $f_X(x)=P(X=x)$ but isn't $P(X=x)=0$

When we define a probability distribution function, we say: $f_X(x)=P(X=x)$ and thats equal to some function such as a gaussian But isn't $P(X=x)=0$ for a continuous random variable $X$. Is it ...
2
votes
4answers
92 views

Statistics: Conditional Probability

$P(A│B)=\frac25$ ,$P(B)=\frac14$, $P(A)=\frac13$. Find $P(A\land B)$ $P(B|A)$ Here is what I did: Part 1. $$P(A\land B) = P(A) \cdot P(B)\\ = \frac13\cdot\frac14=\frac{1}{12}$$ Part 2. ...
0
votes
0answers
24 views

How could an estimator be biased but consistent according to mathematical definition?

According to the definition, an estimator can be biased, if $E_{\theta}[\hat{\theta}]\ne\theta$, with $\theta$ as parameter for a distribution we want to get from samples. While the estimator can be ...
0
votes
3answers
19 views

The probability that the output of the experiment is Y is ___?

Consider the following experiment. Step 1. Flip a fair coin twice. Step 2. If the outcomes are (TAILS, HEADS) then output Y and stop. Step 3. If the outcomes are either (HEADS, HEADS) or (HEADS, ...
2
votes
1answer
33 views

Conditional expectation $E(XY\mid Z)$

I'm trying to solve the following problem: let $X$ and $Y$ be 2 independent standard normal random variables and let be $Z=X+Y$. Calculate $E(XY\mid Z)$. I tried many approaches, but without getting ...
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votes
1answer
54 views

Expected Value of a in a randomly chosen Rectangle

There is a N×M grid. Each square in the grid either has or does not have a mango tree. For example, suppose the field looks as follows. We Know That there are K Mango Tree. ...
0
votes
1answer
28 views

Probabilities for rolling multiple dice and getting one number or greater

I am interesting in producing a table of probabilities for dice rolls. These are standard 6 sided dice. What is the probability that for rolling X dice, Y dice will roll (hit) at least number Z or ...
0
votes
1answer
32 views

Probability of getting a number in a sudoku box when two numbers are already fixed.

Imagine a sudoku box, I named the rows by alphabets like $a,b,c,d...$ And the columns as $1,2,3,4...$ If two numbers were already filled,i,e at $(a,1)$ there is '$1$' and at $(b,2)$ there is '$2$' ...
1
vote
3answers
55 views

Does $n/\Sigma_{i=0}^n(1/X_i)$ converge to $0$ in probability for $X_i$ iid standard uniformly random variables?

Suppose $X_i \sim\operatorname{uniform}[0,1]$ and that they are iid. Does $n/\Sigma_{i=0}^n(1/X_i)$ converge to $0$ in probability? A simulation seems to indicate that it does. But as the expected ...
0
votes
1answer
11 views

Proof of Probability for Exponential RV's where $ x_1 < x_2$

I have a theorem in my notes that says that if X is an exponential RV, $X$ ~ $exp(\lambda_{1}+\lambda_{2})$, $$Pr(x_{1} < x_{2}) = \frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}}$$ I was too far ...
0
votes
1answer
11 views

Determine the transition probability matrix, Simple Insurance Company…

The Simple Insurance Company starts at time $0$ with a surplus of $3$. At the beginning of every year, it collects a premium of $2$. Every year, it pays a random claim amount as shown: $0$ with ...
0
votes
0answers
21 views

3 card poker ante bet computation [on hold]

On the 3 card poker, I have came up on the possible combinations that can be generated from a 52 card of deck. and it was correct. now my problem was how they came up with the values of the Ante ...
1
vote
1answer
28 views

Find the joint probability density given the support set

Suppose that the support set of $(X,Y)$ is $$S_{X,Y}=\{(x,y)\in\mathbb{R}^2: x \geq 0 \text{ and } 0 \leq y \leq e^{-x/3}\}$$ $(X,Y)$ is uniformly distributed on $S_{X,Y}$. a) Find the joint ...
1
vote
1answer
45 views

How many ways are there to arrange these letters? [on hold]

So I've been working out how many ways there are to arrange the letters of probabilistic. I came up with $518918400$ ways. The next thing I want to figure out is out of those ways, how many of them ...
3
votes
1answer
55 views

Almost sure bounded imply finite expectation?

Suppose that the random variable $X$ is $\mid X \mid<M$ almost surely, for some constant $M<\infty.$ Then can we say that $E(X)<C$ for some constant $C<\infty$? If the expectation is not ...
2
votes
1answer
19 views

Exponential law with both positive and negative values

The exponential law with density $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$ and $f(x)=0$ for $x < 0$, is well-known. What's the name of the distribution which has $$f(x) = \frac{1}{2} ...
0
votes
3answers
35 views

Probability of an odd amount of sixes when rolling a 6-sided die 10 times.

Rolling a fair die 10 times, what is the probability it will give an odd amount of sixes? So the outcomes I'm interested in are: 1 six in 10 rolls or 3 sixes in 10 rolls or 5 sixes in 10 ...
2
votes
2answers
40 views

AM-GM Inequality Confusing

Here is something that I find hard to make sense of. Suppose $X_1, X_2, ..., X_n$ are independent draws from some distribution. By AM-GM inequality, we have: $$ \left( X_1 X_2 .. X_n ...
2
votes
2answers
49 views

Proving a Trick to More Quickly Calculate N-Step Transition Probabilities

So, I have been working on a homework problem all day that asks me to prove that: $P^n= \Pi +Q^n$ where P is the transition matrix of a finite-state regular Markov Chain, $\Pi$ is a matrix whose rows ...
0
votes
1answer
22 views

probability question: word game

Suppose i have a bag with all letters of the alfabet. I pick $1$ letter and i put it back. I pick like this 20 letters (so duplicates are allowed). I need to calculate the change that i can form a ...
3
votes
3answers
32 views

Consecutive strings of heads problem

So the question asks: We toss a fair coin $n$ times and record the outcome as a sequence of H and T. We say that there is a run of heads if there is a consecutive string H...H which starts either at ...
0
votes
0answers
9 views

Continuous time markov process

If a stochastic time X(t) t $\ge$ 0 is a Markov Process defined on a finite space, then must it be a jump process?
0
votes
1answer
15 views

Expected value prove problem

So the question asks: Let Y ≥ 0 be a non-negative random variable. Prove that that for any $t > 0$, P (Y ≥ t) ≤ E [Y ]/t So so ...
0
votes
0answers
18 views

Expectation of the product of a random variable squared and its third derivative

Here is how the problem is posed. Show: $$ \left \langle u^2\frac{\mathrm{d}^3 u}{\mathrm{d} t^3} \right \rangle = -2\left \langle u\dot{u}\ddot{u} \right \rangle =2\left \langle \left ( \dot{u} ...
2
votes
3answers
57 views

How do I calculate dice with addition and subtraction based on dice rolls?

I am trying to figure out how to calculate results on a group of dice where some results are positive and others are negative. Example: I roll a group of dice that are fair and six-sided. Each roll ...
0
votes
1answer
37 views

Convergence in Distribution to the normal distribution.

let $ X_1,X_2+,...,$ be independent and identically distributed random variables with Poisson Distribution, does $$ \frac{1}{\sqrt{n}}\sum_{i=1}^n(X_{2i-1} - X_{2i})$$ Converge in distribution to ...
0
votes
1answer
41 views

another follow up question: modeling with exponential distributions

This a follow up question to the previous two: modeling with exponential distributions a follow up question about modeling with exponential distributions I'm trying to do (c). Denote the ...
0
votes
0answers
24 views

Probability to reach final state

Let $~~m,n>0~~$ be some positive integers. We have some system of states. Each state is pair $~~(i,k)~~$ where $~~0\leq i \leq m~~$ and $~~0\leq k \leq n~~$. Starting state is $~~(m,n)~~$. For ...
0
votes
1answer
20 views

CDF of the highest result of multiple unform random variables.

Say I have multiple uniform random variables. I want to know the CDF for selecting the highest result of all the variables. As an example, say I have 3 uniform random variables from [0, 100). Using a ...
0
votes
1answer
17 views

density function of $W = X^2$ when $X$ is uniform with disjoint intervals

I'm having some trouble figuring out this (admittedly) very easy problem. Hoping ya'll could help me figure out where I'm going wrong: Let $X$ be uniform on $(-2,1)$ and $(1,2)$ and derive the ...
-2
votes
0answers
49 views

Expected Value of a Mangoes [on hold]

There is a $N \times N$ grid. Each square in the grid either has or does not have a mango tree. For example, suppose the field looks as follows: ...