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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25 views

Is there a name for a “trivial” random variable defined on a finite probability space?

Let $(\Omega,2^\Omega,\Bbb P)$ be a finite probability space where $\Omega$ is a finite subset of $\Bbb N$. A "trivial" random variable $X$ st. $X(\omega)=\omega$ for any $\omega \in \Omega$ is ...
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3answers
28 views

{students 1 and 2 are in different groups} vs {students 1, 2, 3, and 4 are in different groups}

Source: Example 1.11, p 26, *Introduction to Probability (1 Ed, 2002) by Bertsekas, Tsitsiklis. Hereafter abbreviate graduate students to GS and undergraduate students to UG. Example 1.11. A ...
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0answers
23 views

How can you picture Conditional Probability in 3D?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I modified the following's source for concision. $1.$ Now look at ...
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1answer
23 views

How can you picture Conditional Probability in a 2D Venn Diagram?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I pursue only intuition; do not answer with formal proofs. Which ...
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2answers
47 views

How does scaling $\Pr(B|A)$ with $\Pr(A)$ mean multiplying them together?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I modified the following's source for concision. $1.$ Now look at ...
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1answer
16 views

Different sign in solution on probability problem

It is the problem 1.2.2 of Karlin's book Introduction to stochastic modeling: Let $N$ cards carry distinct numbers $x_1, x_2, ..., x_n$. If two cards are drawn at random without replacement show ...
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1answer
25 views

Statistics - Probability

QUESTION If a new type of torch battery has a voltage that is outside certain limits, that battery is characterised as a failure (F); if the battery has a voltage within the prescribed limits, it is ...
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0answers
31 views

Building a “random rectangle”

Adam and Daniel are building a "random rectangle" using a standart pack of cards $(\spadesuit, \heartsuit, \diamondsuit, \clubsuit)$. Adam takes $6$ cards one by one with return, the number of ...
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1answer
42 views

How to calculate $x\%$ chance of success?

Everything I've looked for points to Binomial Distribution, but I have no idea how to use it. Basically, I have $2$ sets of $7$ rolls. Each roll has a specific chance of success. Set 1: ...
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0answers
21 views

Integer and fractional pars of Gaussian random variables

Are there any interesting results known about integer and fractional parts of $\xi \in \mathcal N(0, \sigma^2)$? In particular, I am interested in their expected values and covariance matrix.
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0answers
54 views

Fliping a fair coin $3$ times

Fliping coin $3$ times, let $N$ denote the number of times that we have the same result as the previos flip, and let $i$ be the indicator random variable of the event "not all the $3$ flips are the ...
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2answers
67 views

Prove that if $X$ is subgaussian, then ${\bf E}e^{tX}=1+\sum_{k=1}^{\infty}\frac{t^k}{k!}{\bf E}X^k$

Prove that if $X$ is subgaussian, then $${\bf E}e^{tX}=1+\sum_{k=1}^{\infty}\frac{t^k}{k!}{\bf E}X^k$$ So basically I just need to push the integral through the infinite sum $${\bf ...
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2answers
36 views

What is the intuition for adding vs multiplying probabilities?

Caution: I modified this original answer to simplify the examples. You add probabilities when the events you are thinking about are alternatives (eg: A soccer team scores 0 goals or 1 goal or 2 ...
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1answer
29 views

How do I decide whether an Event is a Conditional Probability or an Intersection?

My question is based on Example 1.9, p 22, *Introduction to Probability (1 Ed, 2002) by Bertsekas, Tsitsiklis. Define the event $A$ = {an aircraft is present} and $R$ = {the radar registers an ...
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1answer
22 views

An example of a reversible but reducible Markov chain

The reversibility of a Markov chain is defined in the following way with some basic propositions. Unfortunately all examples of reversible Markov chains shown in my textbook so far are irreducible, ...
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1answer
25 views

An example of a reducible random walk on groups?

Random walk on group is defined in the following way as a Markov chain. A theorem says the uniform distribution is stationary for all random walk on groups. If the random walk is ...
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1answer
37 views

Maximum likelihood estimator of $\lambda$ and verifying if the estimator is unbiased

$(X_1,...X_n)$ is a random sample extracted from an exponential law of parameter $\lambda$ Calculate the likelihood estimator $\nu$ of $\lambda$. Then, if $n=2$: establish if $\nu$ is a unbiased ...
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0answers
31 views

The patterns of markov chain [closed]

I have one question let $X$ be a Markov chain that could take the values $1,2$ or $3$ with the same probability $1/3$. what is the probability that $(1,2,1)$ pattern occurs sooner than $(2,1,3)$?
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1answer
34 views

Please help me to understand how to read statistical tables

Sorry I never learnt from a professor or class how and now when I look at them I don't know what to do. Here is an example. The Chi Squared table, ...
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0answers
15 views

renewal process,question [closed]

Let $(N_t)_t$ a renewal process. How do you show that $N_t\to+\infty$ with $t\to+\infty$ Thank you.
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1answer
20 views

propertis of Truncated Normal 2 [closed]

If x has a Truncated Normal Distribution (ΞΌ,Οƒ) and $$E(x)=\mu+\sigma \frac{\phi(l)}{1-\Phi(l)}$$ what is the E(|x|)?
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0answers
42 views

What is the probability of a pen touching a bar given that the length of the pen is $10$ cm and the bars are regularly spaced at $15$ cm?

Problem: If a pen of length $10$ cm is thrown out of infinitely large window having vertical bars regularly spaced at $15$ cm, then find the probability that it will touch any of the bars. (Assume ...
4
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1answer
114 views

How do you find the probability of A winning if the probability of getting a favourable outcome in the $r^{th}$ turn is a function of $r$?

Problem: Two players A and B are playing snake and ladder. A is at 99 and he needs 1 to win in rolling of a dice. However, he is always allowed to re-throw the dice if 6 appears. What is the ...
4
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1answer
72 views

From marginal distribution to joint distribution

Consider two sequences of real-valued random variables, $\{X_n\}_n$ and $\{T_n\}_n$. Let $\rightarrow_d$ denote convergence in distribution. Assume (1) $X_n\rightarrow_d L$ as $n\rightarrow \infty$, ...
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0answers
27 views

Existence of a solution of a limit of Fixed point equations

I am considering a setting where I am given an iid sample of symmetric positive matrices $\{S_i\}$, $i=1,\dots, n$, of a matrix valued random variable $S$ with distribution $F$. The support of $F$ is ...
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2answers
51 views

Combination - Distribution of gifts

Seven different type of gifts are to be distributed among 10 children.Every kind of gift must be at least given to one child. Then, how many combinations do we have? Note:You have A, A, A.... ...
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1answer
77 views

Probability of winning tornament

"Consider a knockout tournament which has reached the semifinal stage. Based on their current form, their scores and ranks are given below. It is believed that whenever two teams face each other, the ...
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0answers
29 views

Is the following modification of a martingale still martingale? [on hold]

I have a following question. Let $Z$ be a Geometric Brownian motion, $\frac{dZ(t)}{Z(t)} = \omega dt + \sigma dW(t) $ For $\omega = -\frac{1}{2}\sigma^{2}$ one can proof that $Z$ is a martingale. ...
3
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1answer
53 views

The partial derivative of a characteristic function (exercise).

Assume that you have a probability space $(\Omega, \mathcal{F},P)$ and a random varaible $X: (\Omega, \mathcal{F})\rightarrow(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$. Define the characteristic ...
2
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2answers
96 views

Which distribution function for diseases

I am wondering, what would be a good method for choosing a suitable probability distribution to fit a certain criteria. For example, I am wanting to choose a suitable distribution, and specify the ...
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1answer
86 views

Probability distribution for score of dice game?

In the game, ten $D_{20}$ (twenty-faced) dice are rolled. If any of the dice are $1$, you remove one of the "$1$", and get a point. Of the remaining dice, if any are $\le 2$, you remove one of them ...
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1answer
32 views

Expected number of rolls in craps conditional on the fact that the house wins.

I'm currently trying to find the expected number of rolls in a craps game given that the house wins. In a craps game, two fair 6 sided die are rolled. The sum of both is then calculated. If the sum is ...
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1answer
25 views

marginal probability density function for variables with uneven range

I'm working through some mock exam questions in preparation for an upcoming exam, and I'm stumped on this one. given that $ f(x,y) = \frac 3 4 y, \quad x \in ]-1,1[, \quad 0 \le y \le x+1$, I'm to ...
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0answers
52 views

Calculate $\text{Var} [-3Z ] $

$X$ and $Y$ are independent random variables. $X$ : Poisson with parameter $1$ $Y$ : Bernoulli with parameter $\frac{1}{3}$ $Z=X+Y$ Calculate $\text{Var}[-3Z]$. I consider that ...
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1answer
29 views

Probability to fit $1950$ items in a box that hold $1880$. Risk $5$%

If I buy $1950$ plates to fill a box that hold $1880$ what is the probability that $1950$ is enough if the risk of dropping a plate is $5$% per plate? The answer is $F_z(-2,81)=0,0025$ I just don't ...
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1answer
49 views

Which matrices give the same probability mass function

In a previous question I asked the following. Consider a fixed (non-random) $3$ by $n$ matrix $M$ whose elements are chosen from $\{-1,1\}$. What is the probability mass function of $Mx$ when $x$ is ...
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0answers
14 views

What is the probability that a player with bankroll of x, flat betting y amount on odds z will bust before he can double his bankroll?

Have found the following question: A player sits down at a roulette table with 20. He bets 1 at a time on either red or black. Either bet pays even money and has a probability of 9/19 of ...
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2answers
45 views

What is the probability that at least $2$ out of $4$ digits in a code is the same?

If I choose $4$ digits for a code randomly out of the digits $0$ to $9$. What is the probability that at least $2$ of these digits are the same? By at least I mean that you have to count with the ...
10
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4answers
419 views

confusion about permutation

$7$ white identical balls and $3$ black identical balls are randomly placed in a row. The probability that no two black balls are together is ? I am getting it as $ \frac{1}{3}$ while the answer in ...
5
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1answer
62 views

Transformed probability distribution function (non-continuous transformation)

Let $$ F_X(x) = \left\{ \begin{array}{ll} \frac{1}{3}e^x & x < 0\\ 1 - \frac{1}{2}e^{-x} & x \geq 0 \end{array} \right . $$ What is the distribution of $Y = F(X)$? I have a hard time ...
0
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1answer
38 views

How to get expectation of minimum or maximum of exponential distribution

We have $n$ i.i.d. samples $X_1,X_2,\dots,X_n$, which have exponential probability density with mean $\mu$, so pdf is $f(x)=(1/\mu)e^{-x/\mu}$ for $x>0$ and $0$ otherwise. Now how to calculate the ...
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3answers
45 views

Palindromic coin toss sequence

A fair coin is tossed 8 times then find the probability that resulting sequence of heads and tails looks the same when viewed from beginning or from the end? How to approach this question because ...
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2answers
64 views

Is there an intuitive way of viewing the Law of Total Expectation $\mathbb{E}\big[\mathbb{E}[X|Y]\big]=\mathbb{E}[X]?$

Law of total expectation If $\mathbb{E}\big[|X|\big]$ finite then for any $Y,\;\mathbb{E}\big[\mathbb{E}[X\mid Y]\big]=\mathbb{E}[X]$ I remember reading this for the first time and ...
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6answers
5k views

If we randomly select 25 integers between 1 and 100, how many consecutive integers should we expect?

Question: Suppose we have one hundred seats, numbered 1 through 100. We randomly select 25 of these seats. What is the expected number of selected pairs of seats that are consecutive? (To clarify: we ...
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0answers
29 views

How should I approach this Poisson distribution problem?

Show that the Poisson probabilities $ π‘ƒπœ‡(π‘˜)$ satisfy the recurrence relation $π‘ƒπœ‡(π‘˜)=\fracΞΌkπ‘ƒπœ‡(π‘˜βˆ’1)$ and hence determine the values k, for which the terms $π‘ƒπœ‡(π‘˜)$ reach their maximum for ...
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0answers
21 views

Convergence in Distribution of Minimum

Suppose $\{X_n\}$ are iid non-negative random variables with common density $f(x)$ satisfying $$ \lim_{t\to 0}f(t) > 0. $$ Show that $n\bigwedge_{i}X_i$ has a limit distribution. Any ideas on ...
0
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1answer
24 views

Calculating probability from a summation of the Negative Binomial Distribution

I am trying to understand an example of calculating $P(X>75)$ for a random variable $X$ when you have the formula (from the Negative Binomial Distribution for this example): $$f(x)= P(X=x)= {x+4 ...
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2answers
37 views

Expectation of a continuous random variable with probability one at a given point.

$$f_{X}(x) = \begin{cases}\frac{x+1}{a+1} & -1 \leq x < a \\ 1 & x=a \\ \frac{x-1}{a-1} & a < x \leq 1 \\ 0 & \text{otherwise}\end{cases}$$ Calculate $E(X)$ I know typically ...
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1answer
40 views

Show that we always have $Y + Z = X + 4$.

Let $X$ be a Geometric random variable with parameter $p =\frac{1}{2}$. We define another random variable $Y$ in terms of $X$ as follows. $Y = \min\{X,4\}$ Here $\min\{X,4\}$ is the minimum between ...
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2answers
59 views

Two points are selected on a straight line of length 'a' units at random

If two points are selected on a straight line of length 'a' units at random, then what is the probability that none of the three line segments formed by the two random points has length less than a/4. ...