# 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.

67,666 questions
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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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### 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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### 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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### $P(X = n) = pq^{n-1}$ 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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### 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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### 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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### 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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### 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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### 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: ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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-...
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 ...