Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
4 views

How to find earnings, expected earnings and stock?

I am doing a study problem and I would like to know if my answers look fine. X is a random variable that represents the demand for a product with the following density function: $$f(x) = \frac1{e^x} ...
0
votes
2answers
14 views

In how many ways can the votes of n voters be split among k candidates?

Suppose there are n voters and k candidates. In how many different ways can the vote be split among the candidates? To be clear,...
0
votes
1answer
31 views

Lottery odds: Are some tickets more valuable than others?

Let's say the odds of winning a scratch-off lottery game are exactly 1 in 1000, and I purchase 100 tickets at a local convenience store. The company that makes the lottery tickets always creates them ...
0
votes
1answer
18 views

Finding a joint CDF F

Given the joint density of two random variables $X$ and $Y$, $f_{XY}(x,y)=2e^{-(x+y)}$ for $0<x<y$ How do I find the joint CDF ? I know it'll be: $F_{XY}(x,y)=\int\int_R f_{XY}(x,y)=\int\...
0
votes
2answers
20 views

25% chance occurring 7/11 times.

If an event has a probability of happening 25% of the time. How do you calculate the chances of this happening 7 out of 11 times. If A is 25% and B is 75%. What is the probability of A occurring 7 ...
0
votes
0answers
19 views

Poker chance calculation with unknown cards

I'm trying to recreate a poker chance calculation like they do on television or poker broadcasts, but without knowing the opponent cards. On the internet you can find probabilities of having this kind ...
0
votes
1answer
19 views

Probability for sampling at least one member of a set from a larger set

Let's assume the following: Population size: N Individuals with a particular feature: x% of N Sample size: y% of N What's the chance that we get at least one ...
1
vote
1answer
25 views

Variance of the sum of n-independent random variables

Let $X_{1}, X_{2},\ldots,X_{n}$ be $n$-independent non-identical random variables. Here I define $a = 5$ and $E\left(X_{i}\right) = a/n,\quad 1 \leq i \leq n$. Then I simulate the Summation process of ...
0
votes
1answer
15 views

Finding the probability of a joint density

Given the joint density of two random variables $X$ and $Y$, $f_{XY}(x,y)=2e^{-(x+y)}$ for $0<x<y$ How do I compute $P(Y<1|X=1)$? I know the conditional probability formula is: $P(Y<1|...
0
votes
2answers
25 views

Finding covariance given mean and variance of both X and Y

Say the distribution of $X$ is known, and the expected value and variance of $Y$ is known. Don't assume independence. Is this information enough to give the covariance of $X$ and $Y$? I am only ...
0
votes
0answers
17 views

Using mirrored sample data to improve estimates

I will ask my question through an example game: Each round we are given a blue coin and a red coin. We are either given a pair of fair coins or a pair of biased coins (these four coins are the set of ...
0
votes
2answers
14 views

How can you decide the winning percentage of a player in a game based on PMFs for each player?

In a two player game, each player is trying to score the higher number of points. We can assume that the probability that a player scores $n$ points is independent of how many points his opponent ...
5
votes
3answers
21 views

Multi dice probability [on hold]

Any number on a dice (2 for example) has a 1 in 6 chance of showing. But if I throw 6 dice all at once, is it still a 1 in 6 chance a 2 will be among the results? If I threw 100 dice all at once it ...
0
votes
3answers
52 views

What is wrong in this approach?

I was trying to solve :(sixteen players $s_1, s_2, s_3, \ldots, s_{16}$ playing a tournament are divided into eight pairs at random) Sixteen players $s_1, s_2, s_3, \ldots, s_{16}$ playing a ...
0
votes
0answers
18 views

Convergence of stopping times

Let $t > 0$ and let $\{ T_k \}_{k \geq 1}$ be a sequence of non-decreasing stopping times with $T_{k} \uparrow t$. Let $(X_s)_{s \geq 0}$ be some stochastic process. I am looking for a result of ...
0
votes
1answer
13 views

Law of total variance and covariance given X and Y are normal

I have a problem which asks me to find $\Bbb E[Y]$ and $Var(Y)$ given that $Y\text{~}Normal(x,1)$ conditional on $X=x$. $X$ is standard normal. So I have worked out that $\Bbb E[Y]=0$ using the law of ...
0
votes
1answer
7 views

Marginalizing by sampling from the joint distribution

For two random variables $x$ and $y$, if I can sample from the joint distribution $p(x, y)$, I can obtain samples from the marginal $p(x)$ by sampling from the joint distribution and ignoring the ...
1
vote
2answers
31 views

How do I calculate the conditional expectation?

How do I find a conditional expectation, $E(Y|X)$ for: $f_{XY}(x,y)=2e^{-(x+y)}$ for $0<x<y$. I researched that the formula for a conditional expectation is: $E(Y|X) = \Sigma yf_{Y|X}(y|x)$ ...
1
vote
2answers
26 views

Distribution function for $f: \mathbb{R}^m \to \mathbb{R}^n$ where $m < n$

Suppose $X$ is a random variable in $\mathbb{R}^n$ and $Y = f(X)$ where $f : \mathbb{R}^n \to \mathbb{R}^n$. $Y$ is a random variable as well and the probability distributions for $X$ and $Y$ are ...
0
votes
1answer
13 views

Finding the conditional density

How do I find a conditional density of Y, given X = x and $f_{Y|X}(y|x)$? $f_{XY}(x,y)=2e^{-(x+y)}$ for $0<x<y$. I know: $f_{Y|X}(y|x)=\frac{f(x,y)}{f_{X}(x)}$? And calculated $f_{X}(x)$ to ...
0
votes
1answer
13 views

Deriving a probability from a moment-generating function

If the MGF of $X$ is $\beta^te^{t^2}$, for some $\beta>0$, find $\Bbb P (X>\text{log}_e(\beta))$. So far, I can see that the MGF of $X$ is similar to that of a standard normal distribution; ...
0
votes
0answers
13 views

Understand a probability distribution

I have a message transmision channel with input: a vector $x\in F_2^n$ and output: a vector $y\in F_2^n$ such that $y\neq x$. I call $E$ the set of positions where an error was transmitted, and $P$ ...
1
vote
1answer
18 views

MLE of θ from a normal distribution

I have the pdf of a normal distribution $$ f(y;θ)= \frac {1}{y\sqrt{2π\theta}}exp (-{\frac {(\log y)^2}{ 2θ}})$$ for $y>0$ and $\theta>0$ and $f(y; θ) = 0$ otherwise. and assuming that $E(\log ...
0
votes
1answer
20 views

Is it $g(\alpha X+\beta Y) =g(\alpha)X+g(\beta)Y$ right?

Consider an example of Jensen's inequality in Probability. Let $X(\omega)=x\mathbf{1}_{A}(\omega)+y\mathbb{1}_{A^{c}}(\omega)$, for some $x, y\in \mathbb{R}$ such that $\mathbb{P}(A)=\lambda$. We have ...
0
votes
1answer
22 views

Computing Probabilities

Given the joint density of two random variables $X$ and $Y$, $f_{XY}(x,y)=2e^{-(x+y)}$ for $0<x<y$ How do I compute $P(Y<1|X<1)$?
0
votes
1answer
16 views

Finding a marginal CDF

I am stuck finding a marginal CDF of X given the random variables $X$ and $Y$ with the joint density: $f_{XY}(x,y)=2e^{-(x+y)}$ for $0<x<y$
0
votes
1answer
21 views

Difference between unnormalized and normalized probability.

I have an equation : e^y, which is called unnormalized probability. And another equation : e^y/sum(e^y), which is called normalized probability. I am not getting the difference between the terms they ...
1
vote
1answer
24 views

Deriving MLE of $\mu$ in Multivariate Gaussian Distribution

Let's consider the gaussian distribution which the covariance matrix is known, suppose we have $D$-dimensional $N$ data $y_n \in \mathbb{R}^D$. And the likelihood function of $\mu$ is $$L(\mu)=\Pi_{n=...
1
vote
0answers
13 views

Concentration inequality for the maximum of non-iid Gaussians (i.e. general case)

I am looking for an upper tail bound on the maximum component of a Gaussian vector in the general case -- i.e., arbitrary mean and full covariance. Specifically, let $x \sim N(\mu, \Sigma)$ where $\...
0
votes
1answer
15 views

Finding negative 2nd moment of gamma distribution

Given a gamma distribution with shape $\alpha=2$ and rate $\lambda=10$, I was first asked to find an expression for $\Bbb E[X^k] \ \forall \ k \in \Bbb N$. Directly computing this, I got $$\Bbb E[X^...
2
votes
1answer
26 views

Characterization of the geometric distribution

$X,Y$ are i.i.d. random variables with mean $\mu$ , and taking values in {$0,1,2,...$}.Suppose for all $m \ge 0$, $P(X=k|X+Y=m)=\frac{1}{m+1}$ , $k=0,1,...m$. Find the distribution of $X$ in terms of $...
0
votes
0answers
24 views

“Ride the Bus” optimal game strategy

Ride the bus is a drinking game that involves making a set of choices correctly in a row to win and stop playing. The player who is riding the bus must complete all four in a row. A random card is ...
0
votes
0answers
39 views

A Probability Thought Experiment

Scenario: Lets say you have 100 trillion unique locks and their corresponding 100 trillion unique keys. You scramble them up, and then place all the locks and all the keys in two seperate boxes. ...
1
vote
0answers
15 views

Finding distance distribution between a random point and a known point in a circle

Here is a circle with radius R. Its is known that any random point in a circle has distribution $f(r)=\dfrac{2r}{R^2}$. I want to know the distance between any chosen point on the circle and a fixed ...
0
votes
0answers
20 views

Probability Density Function - Possible Values to produce a valid PDF

The question is : $f(t) = \frac{5z}{16x}t^2 + z\frac{x-y}{4x}t$ ($x,y,t \in \mathbb{R}$) $(0 < t < 2)$ Express $z$ in term of $x$ and $y$, given $ t$ a random continuous variable? What are ...
0
votes
1answer
20 views

Recurrence formula for the moments / product moments of some order statistics

I am just interested in $E[L_n], E[U_n], E[L_n U_n], E[L_n^2]$ and $E[U_n^2]$ where $L_n =\min(X_1,\cdots,X_n)$ and $U_n=\max(X_1,\cdots,X_n)$. The $X_k$'s are i.i.d. In fact, I am only interested in $...
0
votes
1answer
17 views

Given $P[S|A\cap B]$ and $P[S|A\cap B']$ can we find P[S|A]?

Is there a theorem to or formula to solve the above probability question? Here are the specifics: $P[S]=0.36$ $P[A\cap B]=0.56$ $P[A\cap B']=0.21$ $P[S|A\cap B]=0.6$ $P[S|A\cap B']=0.45$ $P[S|A]...
0
votes
1answer
25 views

What is the probability that a random walk starting at 0 will reach +2 in 2 steps, 3 step, 4 steps, etc.? [duplicate]

The random walk I am referring to is a symmetric, unbiased, 1D random walk. In an answer given in the link below, the probabilities are given for S1, but I am trying to find out what it is for S2, ...
-1
votes
1answer
26 views

How could i show(prove) this exercise of probability about moment generating function [on hold]

I have to show this exercise of probability but i don't have idea how prove that: using the moment generating function, prove that if X is distribuited $\mathcal N(\mu,\sigma^2)$ then $2\mu - X$ ...
1
vote
1answer
60 views

Finding $E[X]$ from the joint density function of $X$ and $Y$.

The joint density function of $X$ and $Y$ is given by $$f(x,y)=\frac{1}{y}e^{-(y+\frac{x}{y})},\quad x>0,y>0.$$ Find $E[X]$, $E[Y]$ and $Cov\left(X,Y\right)$. Calculating $E[Y]$ was easy for me....
1
vote
2answers
30 views

why is the probability of pulling blackjack from a full deck $\frac{4\cdot 16\cdot 2}{52 \cdot 51}$?

I understand that the 4 in the numerator is for the aces and the 16 is for drawing a king, queen, jack or 10. (52*51) in the denominator is all possible permutations of draws. I'm not really sure why ...
1
vote
2answers
32 views

Let $\bar X_n$ be the sample mean. What is the accurate rate of $\bar X_n-\mu$ convergence to $0$,

Suppose $X\sim N(\mu,\sigma^2)$ and $X_1,\cdots,X_n$ are samples from $X$. Let $\bar X_n=\frac1n\sum_{i=1}^nX_i$. Then it is well known that $$\bar X_n\overset{p}\to\mu\qquad\qquad(1)$$ and $$\sqrt{n}(...
2
votes
3answers
61 views

Why is the probability that a continuous random variable takes any one specific value equal to 0? [duplicate]

What would the intuitive explanation be? Would it be because there are infinately many values and so the probability of any specific value is infinitely small hence we say 'close enough' to 0?
0
votes
1answer
28 views

What is the probability Event $B$ Occurs and $A$ does not [on hold]

I am given $P(A)=0.40$ $P(B)=0.30$ and $P(B|A)=0.4$
1
vote
2answers
49 views

Balls in Urn question

There are 10 red balls, 20 green balls, 30 blue balls in an urn. We take them all out one by one without replacement. What is the probability that by the time we take all the blue balls out there is ...
1
vote
1answer
34 views

Probability exercise with normal distribution

So this is a question about Normal Distribution (correct me if I am wrong), but I don't know how to solve it as I don't have variance and I'm not sure about mean. Help is highly appreciated. Assume ...
0
votes
0answers
14 views

Mutual information preserves order under scaling single column?

The problem arises when we tried to use mutual information to evaluate importance of different events to some experiment outcome. The mathematical formulation is as follows. We have two categorical ...
0
votes
3answers
47 views

Probability distribution difficult exercise [duplicate]

I have to solve this exercise and I have no idea how to do it... Help is highly appreciated. We make the following experiment: we ask 2 persons to write one real number from [0, 5] each on a ...
0
votes
1answer
26 views

Distribution of distance from origin for uniformly randomly chosen point in circle

So I think I know how to solve (a) correctly for this problem, but I keep getting answers to (b) that don't integrate to be $1$. I think (c) follows straightforwardly from there so (b) is the big ...
1
vote
0answers
28 views

Card Guessing Game Probabilities

Suppose we have a card game in which, for a standard deck of 52 cards, one card of each suit is selected at random and pulled out of the deck. The remaining 48 cards are shuffled together, and laid ...