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Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

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Shortcut to finding the distribution of a specific random variable

Question: A dice is rolled 3 times. Let X denote the maximum of the three values rolled. What is the distribution of X (that is, P[X = x] for x = 1,2,3,4,6)? You can leave your final answer in terms ...
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0answers
21 views

Calculate Mean from the Moment Generating Function ( m.g.f / mgf) of Y = 2X +3

The question is as follows: No calculators. Let X be a random variable with moment generating function $M_{x}(t) = \frac{e^{(e^t-1)}}{2e^{-t} -1} \;\;\;\;\;for \;\; \;t<log(2)$ given Y = 2X +3 ...
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1answer
42 views

Prove $\lim_{x\to\infty} x . [1 - F(x)] = 0$ [duplicate]

I have no idea how to proceed with proving this. If $X$ is a continuous random variable, $P(X > 0) = 1$, $E(X)$ is defined and $F(x)$ is the CDF, then prove $\lim_{x\to\infty} x . [1 - F(x)] = 0$
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1answer
65 views

$P(X = n) = pq^{n-1}$ where p, q > 0 and p + q = 1. Find Var(X) using generating function.

$P(X = n) = pq^{n-1}$ where $p, q > 0$ and $p + q = 1$. Find ${\tt Var}(X)$ using generating function. First I found $E(X)$: $$\sum_{n=1}^\infty q^n = 1/(1-q) - 1 = q/(1-q)$$ then differentiate ...
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2answers
16 views

Density function of an exponential distribution

Let $X$ be a random variable with an exponential distribution with $\lambda=1$ and $Y=2X$. What is the density function of $f_y$? I know that $$f_x =\begin{cases} e^{-x} & 0\leq x\leq\...
1
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1answer
41 views

Asymptotically tight bounds $P \left[1-\frac{1}{n} \le \frac{\sum_{i=1}^n Z_i^2 }{n} \le 1+\frac{1}{n} \right]$

I am looking for assymptotically tight bounds on \begin{align} P \left[1-\frac{1}{n} \le \frac{\sum_{i=1}^n Z_i^2 }{n} \le 1+\frac{1}{n} \right]=P \left[ \left| E[Z^2] - \frac{\sum_{i=1}^n Z_i^2 }{n} ...
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0answers
25 views

conditional expected value of mean preserving spread

My first post on here, and my math skills are more than a little rusty. I have a simple question for you: assume $Y$ is a mean preserving spread of $X$. I'm trying to find a general proof for the ...
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0answers
33 views

Reverse engineering distributions

Suppose I am given a measurable function $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and a probability distribution $\mathbb{P}$ on the Borel or Lebesgue sigma algebra of $\mathbb{R}^n$. Assume that the ...
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1answer
11 views

Does it make sense to compare coefficient of variance between samples with different sample size?

I have two samples with different sample sizes. The difference is quite large: one has sample size of 10 and one has sample size of 200. Two samples are same type of data but are collected from two ...
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2answers
29 views

Coin game between A and B

Consider the following simple game. In a single round, Player A tosses a fair coin, and then Player B tosses a fair coin. Two rounds are played. The winner is the player with the larger number of ...
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0answers
37 views
+50

Prove periodicity is a class property

Prove that if state $i$ in a class has period $p$ then all states in that class have period $p$. The proof is given on this answer is this: One way to define the period of state $i$ is as the ...
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4answers
52 views

Why the statement: if $P(A|B) > P(A)$ then $P(B|A)>P(B)$ intuitively makes sense?

I know how to prove the following statement but I can't understand why this makes sense intuitively: If $P(A|B)>P(A),$ then $P(B|A)>P(B)$ I would appreciate it if someone can explain it ...
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0answers
18 views

Is a Spindown d20 still fair?

Is a Spindown d20 (numbers arranged consecutively around the instead of 21-sums arrangement) still fair to roll, given enough dice action to eliminate human variables, compared to a standard d20?
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0answers
31 views

Probability of obtuse triangles formed on a circle

There are 16 equally spaced points on the circumference of a circle. If 3 points out of these 16 points are selected randomly, What is the probability that they will form an obtuse angled triangle?
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2answers
44 views

Why is $ce^λ=1$ equal to $c=e^{-λ}$? [on hold]

Why is $ce^λ=1$ equal to $c=e^{-λ}$?
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1answer
24 views

Roll n 3-sided dice (ABC). What's the probability of at least one A and two Bs?

Suppose you have n three-sided dice, with sides labelled A, B, C. What is the probability of getting ABB among your dice (i.e. at least one A, at least two Bs)? Order is not important. (By ...
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0answers
20 views

Population Standard Deviations [on hold]

An individual checking account at a major UK bank costs between $\$350$ to $\$450$ per year. Suppose the current average cost of all checking accounts at major UK bank is $\$400$ per year with ...
1
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1answer
16 views

Probability using binomial distribution

I am trying to le-learn probability theory and I am solving the following problem: A probability that a manufactured device has $3\%$ or more deffects is $p=0.02$. If a company buys $5$ devices, ...
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1answer
19 views

Probability that a die game ends on an even throw (Linear Solution)

I've researched solutions for this problem seeing as I couldn't seem to solve it on my own, and there's one in this very forum that albeit looking the most simple, I can't fully understand. Here is ...
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0answers
34 views

What is the expected value: [on hold]

How would I calculate the expected value of: There is a game involving opening doors. There are 10 doors and 3 contain normal balls while one contains a gold ball. One gold ball is worth 3 points ...
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2answers
30 views

Probability of rolling a die twice

I have a statistics question. Please let me know if I am on the right track. If you throw a die for two times, what is the probability that you will get a three on the first throw or a three on ...
2
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1answer
39 views

Multiplicative Chernoff Bound for coins

We flip 1000 fair coins and mark the results with $X_1, . . . , X_{1000}$. A pair of adjacent coins are $X_i,X_{i+1}$ coins - i count $X_{1000}$ and $X_1$ as adjacent. We refer to the number of pairs ...
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3answers
30 views

Find probability equation when there is two probabilities in the decision tree switching around

The problem is like this: The probability of success at 1st time is 0.625. From the second time, if it is previously successful, the probability of success would become 0.83. If it is previously ...
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0answers
40 views

What is the probability that he or she spent at least 1000 dollars?

In a shop,20% of the customers spend at least 1000 dollars.Among those spending at least 1000 dollars, 30% pay by credit card,while among those spending less that 1000 dollars, only 10% pay by credit ...
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1answer
17 views

Dual Bayesian Interpretation

I am reading a book which says there are two ways of interpreting Bayes. “In the Bayesian approach, parameters can be viewed from two perspectives. Either we view the parameters as truly varying,...
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1answer
12 views

Probability norm less than threshold in unit ball

From exercise 2.4 in Elements of statistical learning, studying this solution : http://tullo.ch/static/ESL-Solutions.pdf Points $x_{i}, i=1..N$ are uniformly distributed in a p-dimensional unit ball ...
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0answers
27 views

Proof finite dimensional equaltiy implies equality on interval

Consider the stochastic process $\{X(t)\}_{t\in T}$ and certain specific operations $g$ and $f$. If $T$ is of finite dimension, i.e. $T=\{t_1,\ldots,t_k\}$, with $t_i\in 0,\infty)$ then I have been ...
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2answers
40 views

proof that $Y=μ+σX$ if X∼N(0,1),

I want to proof that If$$X∼N(0,1)$$, then$$Y=μ+σX$$has the normal distribution with mean $μ$ and variance $σ^2$. I searched it before, but I don't understand why I have to calculate the probability ...
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1answer
39 views

Bayes' Theorem with Multiple Tests

I'm having difficulties with this problem: Suppose you have an entire city afflicted with four distinct and exclusive diseases and a laboratory is assigned to test which disease each citizen has. ...
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2answers
28 views

Comparing two normal distributions

Given a normal distribution $X$~$N(60,9^2)$ with a random variable $A$ and a normal distribution $Y$~$N(50,7^2)$ with a random variable $B$, how do I go about finding the probability $P(B>A)$? (...
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1answer
28 views

A stopped process is adapted

I am trying to understand the proof of Theorem 2.2.2(Optional Stopping Theorem) in Fleming and Harrington's Counting Processes and Survival Analysis. Let $\{X(t):0\leq t<\infty \}$ be a right-...
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0answers
17 views

How to determine the monotonicity of a chi square distribution random variable over its degree of freedom? [on hold]

How to determine the monotonicity of $\frac{\chi^2_{0.95}(k)}{k}$ ? Here k only belongs to positive integers.
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0answers
17 views

Service queue with expiration

from an Internet of Things setting we have the following question. There is an information service. It has a contract with a customer that it would provide information which is at most N seconds old (...
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1answer
37 views

Actuarial practice exam: Find number of exponential data values above threshold given MLE.

I was going thorough an actuarial exam and came across a problem that I can't figure out. Here is the problem as stated on the practice exam: You are given: $\bullet$ An insurance product with ...
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1answer
19 views

A card is drawn from standard deck of playing cards. What is the probability that it is a)a face and heart card b) face or heart card

I am quite confused about b) part. Whether to use $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ Assuming P(A) is the probability of face and P(B) is the probability of heart or just face probability is $...
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0answers
27 views

Stochastic programming: Is the linear program over the vertices the same as over the simplex?

Suppose we have a random variable $W$ with probability distribution, $\Pr(W = w) = p_w \in [0,1], \quad w \in I = \{1, \ldots n\}$ Consider the maximization problem: $$\max\limits_{w \in I} \...
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1answer
22 views

Urn problems: find mean and variance - stuck

I am stuck in a problem, and I can't think of a next step to find the solution. The question is the following: Suppose an urn has $k$ balls, numbered from $1$ to $k$, $k \in \mathbb{N}$. A sample ...
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1answer
42 views

Finding the probability density function of Z=X+Y

I am stuck with finding the pdf for Z = X+Y, I have the pdf for X and Y. The problem: $f_Y(y) = 1/(b-a) ~~\big[25 \leqslant y \leqslant 35\big]$ $f_X(x) = 0.1 - 0.00667x+1.1\times 10^{-4} x^2$ I ...
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2answers
98 views

Finding the expected number of flips needed for a coin having probability $p$ of landing on heads

A coin, having probability $p$ of landing on heads and probability of $q=1-p$ of landing on tails. It is continuously flipped until at least one head and one tail have been flipped. a) Find the ...
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0answers
36 views

Definition of transient state

Consider the following definition Transient States It is often useful to talk about whether a process entering a state will ever return to this state. Here is one possibility. A state ...
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2answers
28 views

Probability of system failure, as it requires two adjacent component to fail

I am stuck with a problem in probability. It says that the system consists of 16 components distributed in corners and intersections as the following: ...
0
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1answer
26 views

conditional probability with exponential random variables

Radio works with two independent batteries. Each one has exponentially distributed lifetimes with mean $1/\lambda_1$ and $1/\lambda_2$ (years). Radio fails to operate as soon as one of the batteries ...
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1answer
56 views

Understanding definition of Periodicity of Markov chain

Consider the following example that is used to understand the definition of periodicity property. Why does it says that: starting in state $1$, it is possible for the process to enter ...
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1answer
19 views

Need help understanding relationship between the inverse cdfs of a unit normal and unit normal squared aka $\chi_1^2$

I'm having problems understanding this comment I saw in a lecture. With a little thought, you can see that because the graph is “folded over”, the $95$th percentile of the $\chi_1^2$ distribution ...
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1answer
28 views

Stock Probability

This appeared on a sample Quant Exam: A stock has a beta of $2.0$ and a specific daily volatility of $2\%$. Yesterday's closing price was $100$ and today the market goes up by $1\%$. What is the ...
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0answers
17 views

Reference for simple Markov chain construction

Let $M_n$ , $n\in\mathbb{N}_0$ be a Markov chain on a general state space $X$. Fix $m\in \mathbb{N}$. ¨ My question is if there's a name / reference for this trivial Markov chain on $X^m$ defined by ...
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0answers
17 views

Conditional expectation problem with integration limit and not only

We are in the shopping center. Customer is paying with card with chance $60\%$, and witch cash with probability $40\%$. When client is paying with card the time of service has exponential distribution ...
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1answer
53 views
+100

Finding the probability density function of a random variable in two dimensions

Let $(X,Y)$ be a point chosen at random from the triangle $\{x,y:0\leq x\leq y\leq 1\}$. $f_{X,Y}(x,y)=2$ if $(x,y)$ is in the triangle, and it is 0 otherwise. Find the probability density function ...
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0answers
19 views

Finding the joint density distribution of $B(1),B(1)+B(2),2B(3)$ where B is the standard Brownian motion

I am trying to figure of the way to find the joint density function $f_{B(1),B(1)+B(2),2B(3)}(x_1,x_2,x_3)$ of $B(1),B(1)+B(2),2B(3)$ where $B(t)$ is the SBM. I know that $$B(1)=x_1\implies B(2)=x_2-...
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0answers
23 views

Birth/Death Processes with constant departure rate

So, I'm looking at a Birth/Death process $Z$ with an arrival rate of $\frac{1}{n+1}$ and a departure rate of $1$. I'm trying to show that this process is positive recurrent and find the stationary ...