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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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 ...
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3answers
12 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
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1answer
26 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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0answers
22 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. ...
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2answers
13 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 ...
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1answer
30 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 '$2$' ...
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3answers
36 views

$X_i$ ~iid $\operatorname{uniform}[0,1]$ distribution, does $n/\Sigma_{i=0}^n(1/X_i)$ converges to $0$ in probability?

suppose $X_i$ ~iid $\operatorname{uniform}[0,1]$ distribution. My question is does $n/\Sigma_{i=0}^n(1/X_i)$ converges to $0$ in probability? The simulation seems it does. But as the mean of $1/X_i$ ...
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1answer
9 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 ...
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1answer
9 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 ...
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0answers
18 views

3 card poker ante bet computation

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 ...
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1answer
25 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 ...
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1answer
39 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 ...
2
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0answers
26 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 ...
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1answer
16 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} ...
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3answers
33 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
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2answers
34 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 ...
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2answers
35 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 ...
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1answer
20 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
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3answers
28 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 ...
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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?
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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 ...
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0answers
16 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} ...
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1answer
44 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
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1answer
31 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 ...
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0answers
24 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 ...
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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 ...
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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 ...
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0answers
38 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: ...
1
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2answers
15 views

Conditional probability - 8 tosses of a coin

We throw a coin 8 times. What is the probability of getting the same number of heads and tails, if on the last three tosses of a coin we got tails?
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5answers
203 views

Probability and the “out of” thing"

I have quite an odd question: I am not able to fully understand the concept of "out of". If I roll a dice once, from a total of $6$ possible outcomes, I'll get 1. Why does that mean a fraction ...
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0answers
14 views

Poisson process and probability

Let $N_t$ be a Poisson process and $T_{N_t}=X_1+\ldots+X_{N_t}$ where $X_i$ has an exponential law ($E(\lambda)$). Let $A_t=t-T_{N_t-1}$ and $B_t=T_{N_t}-t$. Show that for $x,y,t \geq 0$, $P(B_t \geq ...
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0answers
18 views

its about finding probability given the standard deviation [on hold]

for boys, the average number of absence in the first grade is 15 with a standard deviation of 7;for girls, the average of absence is 10 with standard deviation of 6. in nation wide survey, suppose 100 ...
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0answers
13 views

Combining n simultaneously occuring probabilities of an event occuring into one summative probability

I am a bit lost with regards to the problem described a bit further down, because though many methods to approach it are documented in available literature, the verdict as to which model is the most ...
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1answer
12 views

Is $P(W \geq z |V \geq y)=P(U-V \geq z |V \geq y)=P(U \geq z+y)$ correct?

Let $U,V,W=U-V$ random variables with $z,y \geq 0$ $$P(W \geq z |V \geq y)=P(U-V \geq z |V \geq y)=P(U \geq z+y)$$ Is it correct?
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1answer
11 views

Probability distribution of the difference of two random variables

Let $X,Y,$ and $Z$ be random variables, with $Z=X-Y$ and $z,y \geq 0$ $$P(Z \geq z, Y \geq y)=P(X-Y \geq z, Y \geq y)=P(X \geq y+z)$$ Is that correct? Thank you
0
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0answers
19 views

probability combination

McGyver is faced with the problem of opening a safe with 10 buttons numbered from 0 to 9. The safe can be opened by pressing three buttons, not necessarily distinct, in correct order. Realizing that ...
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0answers
18 views

In a sweepstakes giveaway scenario, how does having 2 chances to win the same prize affect the overall odds?

In a sweepstakes giveaway scenario where total entries are expected to result in final odds of 1:93,150.685 for/against a single entrant (after adjusting for multiple entries) and can be won by either ...
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1answer
20 views

Expected error of simplifying to a geometric distribution

While reading an answer related to solving a problem with a geometric distribution, the following question occurred to me. The answer gives two possibilities for replying the OP's question. In the ...
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0answers
12 views

Maximising returns Limiting Risk

This is probably a simple question/solution, but I'm no math expert. I'm looking into a Facebook group that provides bet to try and get up to 50k, you may have heard of it. The premise being that you ...
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0answers
24 views

Is the density of an absolutely continuous distribution necessarily unique? [on hold]

Is the density of an absolutely continuous distribution necessarily unique?
2
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0answers
38 views

Concentration inequality for sum of squares of i.i.d. sub-exponential random variables?

Suppose $X_1, X_2, \ldots, X_n$ are independent and each has the same distribution with a sub-exponential random variable $X$ (for example, $X$ is the square of a standard normal Gaussian variable). ...
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0answers
30 views

Find the minimum number of tickets to guarantee the win of a n-bit binary lottery?

Here's the problem. I just don't know how to approach it. If the 'one error tolerance' were removed, then this would be a simple binomial distribution problem. But now I can't figure it out. In ...
2
votes
1answer
34 views

probability of not getting same number twice in a row after n die rolls

Having rolled a die $n$ times, I want to determine the probability of not getting any number twice in a row. If I wanted the probability of not getting any number three times in a row, I could use the ...
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votes
1answer
11 views

Is relative entropy with respect to a pmf a continuous function?

Is the relative entropy $D(p || q)$ with a fixed pmf $q$, continuous over $p$, where $p \in \{x \in \mathbb{R}^n: \sum_{i=1}^n x_i = 1 , x_i \geq 0 \}$?
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1answer
26 views

Suppose you draw a five-card hand randomly from the deck and get four cards that that would make a straight if you could replace the fifth card…

Suppose you draw a five-card hand randomly from the deck and get four cards that that would make a straight if you could replace the fifth card. (e.g. J 10 9 8 3 or K 7 6 4 3). If you are allowed to ...
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1answer
26 views

Suppose that after a five-card hand is drawn, the cards in it are put back in the deck and another five-card hand is drawn.

Suppose that after a five-card hand is drawn, the cards in it are put back in the deck and another five-card hand is drawn. a) What is the probability that the two hands have no card in common? b) ...
4
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2answers
82 views

What does $-p \ln p$ mean if p is probability?

In statistical mechanics entropy is defined with the following relation: $$S=-k_B\sum_{i=1}^N p_i\ln p_i,$$ where $p_i$ is probability of occupying $i$th state, and $N$ is number of accessible ...
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1answer
26 views

sum of two Dice game

The question is: You have 2 fairly weighted dice. You and an opponent pick any integer one after the other. If your number is closer to the sum of the faces on the rolled dice, you win. Do you want ...
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0answers
39 views

How does a loaded die affect this probability

Suppose I own five different six-sided dice. Four of the dice are fair dice and they are equally likely show the values $1, 2, 3, 4, 5,$ and $6$. One of the dice is loaded and never shows ...