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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2answers
16 views

Three People Rolling a fair Die.

Three players A, B and C take turns to roll a fair die; they do this in the order ABCABC... (a) Find the probability that, of the three players, A is the first to throw a 6, B is the second, and C is ...
-6
votes
0answers
18 views

Expected Value Given pdf [on hold]

Suppose that fifteen observations are chosen at random from the pdf $ f_Y(y)=3y^2$, 0≤ y ≤1. Let X denote the number that lie in the interval [1/2 , 1]. Find E(X).
0
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0answers
6 views

Matching of points in two discrete linear sequences with potentially missing points

This is a question that I've been thinking about in my research lately. I've gone down the route of a few linear-optimization techniques, but nothing particularly spectacular has come up. Anyway, ...
4
votes
2answers
46 views

Can someone explain what a portfolio is in financial math?

I took mathematical probability last semester and now I am taking financial mathematics, but only probability was a pre requisite for financial math (no finance classes were required). These types of ...
2
votes
1answer
10 views

Moment generating function of sample mean of bernoulli random variables

Let $p \in (0,1)$ and $n \in \mathbb{N}$. We consider a sample of $n$ i.i.d. Bernoulli variables $X_1,\dots,X_n$ with parameter p. Computer $E[e^{\lambda\bar{X_n}}]$ such that $\bar{X_n}= \frac{1}{n} ...
0
votes
0answers
8 views

How to do the inverse problem of the kernel density estimation

Given $x_1, x_2,..x_n ; x_i \in R$ that drawn from an unknown distribution $P(x)$ and a constant $ C$ $ 0 \leq C \leq 1$. Find $x^{*}$ such that $$P(x^{*}) =C$$. We want to use the kernel density ...
1
vote
6answers
43 views

Probability of getting $5$ heads on $10$ (fair) coin flips?

Even before attempting the problem, I immediately defaulted to an answer: $\frac{1}{2}$. I thought that this was a possible answer since the probability of flipping a head on one flip is definitely ...
1
vote
1answer
15 views

If I have that $\limsup_{n}E|X_n|^{r} \leq E|X|^{r}$, is that enough to show that $\{|X_n|^{r}:n\geq 1\}$ is uniformly integrable?

If I have that $\limsup_{n}E|X_n|^{r} \leq E|X|^{r}$, is that enough to show that $\{|X_n|^{r}:n\geq 1\}$ is uniformly integrable? I am not sure here if the limsup condition here is as strong as if I ...
4
votes
2answers
35 views

Probability of island having 8 people born with disease, estimate?

The chances of being born with a certain disease are estimated as $1$ in $1200$. What is a good estimate of the chance that an island with $10000$ inhabitants has precisely $8$ people born with that ...
4
votes
1answer
21 views

What is the probability that these two objects are of the same color?

We have $11$ bins with $10$ objects each. Every object is either black or white, and the $i$th bin ($1 \le i \le 11$) has precisely $(i -1)$ black objects in it. Someone selects, uniformly at random, ...
0
votes
1answer
15 views

Three people have been exposed to a certain illness. Once exposed, a person has a 50-50 chance of actually becoming ill. [on hold]

Three people have been exposed to a certain illness. Once exposed, a person has a 50-50 chance of actually becoming ill. a) What is the probability that exactly one of the people becomes ill? I am a ...
2
votes
1answer
27 views

Are these two events $A$ and $B$ independent?

Abe and Bernard are dealt five cards each from the same $52$ card deck. Let $A$ be the event that Abe gets a flush (five cards of the same suit) and $B$ be the event that Bernard’s five cards are of ...
4
votes
1answer
21 views

$p(X)$, $P(Y)$, $p(Z) > 0$ and every pair of these events is independent, then $p(X \wedge Y \wedge Z) > 0$?

Is the following statement true or not? Let $X$, $Y$, $Z$ be $3$ events in the same sample space such that $p(X)$, $P(Y)$, $p(Z) > 0$ and every pair of these events is independent. Then $p(X ...
-2
votes
0answers
8 views

probability that a customer who purchases up to $5$ songs from $4$ music genres prefers jazz and buys at least $3$ songs [on hold]

Customers can choose from $4$ music genres: jazz, rock, new age, country; and can purchase up to $5$ songs. The events are: $A =$ customer prefers jazz and buys at least $3$ songs $B =$ the customer ...
-2
votes
1answer
18 views

Conditional Independent clarification

Let's say I have $3$ events with probabilities $P(A) = 0.5, P(B) = 0.5$ and $P(C)= 0.5,$ and I need to find if $$P(A \cap B \mid C) = P(A \mid C)P(B \mid C)$$ I am tying to prove this by expanding ...
0
votes
0answers
16 views

integral and probability

Let $N_t$ be a Poisson process and $S_{N_t}=X_1+...+X_{N_t}$. Let $A_t=t-S_{N_t}$ and $B_t=S_{N_t}-t$ 1) Show $P(B_t \geq x \ \text{and}\ A_t \geq y)=\frac{1}{E(X_1)} \int_{x+y}^{\infty} P(X_1 \geq ...
0
votes
2answers
16 views

Conditional probability with dependent events

We have 2 dice. One is fair. The other one lands by the following probabilities: 6: 1/2 5: 1/10 4: 1/10 3: 1/10 2: 1/10 1: 1/10 We roll both dice. What is the ...
1
vote
1answer
17 views

Conditional Probability clarification

Here's a sample problem: Before each workout, I either drink a cup of coffee, a gatorade, or a cup of water. The probability of coffee is P(C) = 0.6, the probability of gator is P(G) = 0.3, the ...
0
votes
0answers
29 views

An urn contains $nr$ balls numbered $1,2..,n$

An urn contains $nr$ balls numbered $1,2..,n$ in such a way that $r$ balls bear the same number $i$ for each $i=1,2,...n$. N balls are drawn at random without replacement. Find the probability that ...
2
votes
1answer
19 views

Win/Lose ratios and selection strategies

Imagine the following scenario: You're on a TCG tournament which allowed you to bring N decks with you. After each game, you might select another deck for your next game. You are allowed to keep ...
1
vote
2answers
17 views

Exponential distributions [on hold]

Good evening to all, I'm so much confused about a question; Assume there is a workshop with two machines. The times until the failures of machines $1$ and $2$ are independent and exponentially ...
0
votes
1answer
19 views

Find the probability of B

Suppose you roll a fair 6-sided dice three times. There are $6^3$ possible outcomes and each is equally likely. Let $A_1, A_2, A_3, A_4, A_5,$ and $A_6$ be the events that the last value is ...
0
votes
1answer
32 views

A coin is tossed $m+n$ times. Find the probability of getting atleast $m$ consecutive heads

A coin is tossed $m+n$ times. Find the probability of getting atleast $m$ consecutive heads I already know that the exact same question has already been answered here But I am trying to solve it ...
2
votes
2answers
22 views

chain rule conditional entropy

I have to prove the chain-rule for conditional entropy. I kept getting stuck on one step, so I looked up a proof and found this: \begin{align}H(Y\mid X)&= \sum_{x\in\mathcal X, y\in\mathcal ...
1
vote
2answers
12 views

Possible orderings when the items are not unique?

First of all, I'm sure this question has been answered somewhere on the web, but I am just starting probability and I don't have the vocabulary to know what to look for, which is why I am asking here. ...
0
votes
0answers
7 views

Is the result of a Monte-Carlo simulation of a continuous function and with continuous input distributions again continuous?

Is the result of a Monte-Carlo simulation of a continuos function and with continuos input distributions again continuous? Suppose, we have a continuos function $f$ and a number of continuous random ...
0
votes
1answer
25 views

Equality in Conditional Jensen's Inequality

Conditonal Jensen's Inequality says that for a convex function $\varphi$, a random variable $X$, and a sub-sigma-field $\mathcal{F}$, $E[\varphi(X)\mid \mathcal{F}] \geq \varphi(E[X\mid ...
0
votes
0answers
18 views

On the probability distribution of iterated permutations

I have this little problem that has been nagging me for a couple of months now. It occurred to me when considering the fairness of card shuffling methods. Here's my best attempt at formalizing it: ...
3
votes
3answers
30 views

What is the probability that a randomly chosen positive three-digit integer is a multiple of $7$?

What is the probability that a randomly chosen positive three-digit integer is a multiple of $7$? Is my answer right?: $$\frac{100}{7} = 14 , \qquad \frac{999}{7} = 142$ Then there are $142 - 14 = ...
1
vote
1answer
19 views

Conditional Probablity for two independent events(Formula)

Let there be two independent events $A$ and $B$. To calculate the probability (for a particular condition) we have two relations. $P(A \cup B)=P(A)+P(B)-P(A \cap B)$. $P(A/B)P(B)=P(A \cap B)$, i.e., ...
0
votes
2answers
21 views

What is the probability or percentage or frequency by which hello line will be printed?

I have a below method which is called every one minute from background thread and that background thread keeps running always. ...
0
votes
0answers
21 views

Destined pair 'guessing' game

n people participate in a game. Before the game the participants are put into random secret 'destined' pairs. Each round the participants pick1 their own pairs and ...
0
votes
0answers
24 views

probability,calculus

Let $N_t$ be a Poisson process and $S_{N_t}=X_1+...+X_{N_t}$. Let $A_t=t-S_{N_t}$ and $B_t=S_{N_t}-t$ 1) Show $P(B_t \geq x \ \text{and}\ A_t \geq y)=\frac{1}{E(X_1)} \int_{x+y}^{\infty} P(X_1 \geq ...
0
votes
3answers
28 views

Is every bounded sequence of random variables in $L^1$ convergent? [on hold]

If $\{X_n\}_{n>0}$ is a bounded sequence of random variables is it true that $E(X_n)$ converges?
0
votes
1answer
33 views

Should I use law of large numbers or Chebyshev inequality?

I think the answer is zero. Can anyone tell me whether I should use Weak Law of Large Numbers or Chebyshev inequality . I just need a hint how to proceed. Is my answer 0 correct? Thanks link to ...
0
votes
1answer
29 views

Need Help with continuous random variable probability problem [on hold]

Suppose that an electric device has a life length $X$ which is considered as random variable with pdf: $f(x)=e^x$, $x>0$. Suppose that the cost of manufacturing one such item is $2$. The ...
2
votes
0answers
42 views

Expectation and Variance of $X/(X+Y+Z)$

I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal ...
0
votes
1answer
28 views

Continuous probability - calculate probability of r.v and distribution function

This is the question: $X$ is a continuous random variable whose probability density function is given by $$f(x)=\begin{cases} \frac{1}{9}x^2 & \text{if $0\leq x \leq 3$}.\\ 0 ...
1
vote
1answer
41 views

How to show a sequence of independent random variables do not almost surely converge by definition?

I have a sequence of independent random variables $X_1, X_2, \ldots$ where $$ X_n = \begin{cases} 1 & \quad \text{with probability} \ 1/n \\ 0 & \quad \text{with ...
1
vote
1answer
29 views

Door Prizes - Probability [on hold]

Joe goes to a party with three friends. There is a drawing for four door prizes. Each person has an equal chance of wining a prize. No one can win more than one prize. If there are totally thirty ...
0
votes
1answer
22 views

How to show convergence in probability by just using the definition?

I have a series of random variables $X_1, X_2, \ldots$ where $$ f(X_n) = \begin{cases} 1/n & \quad \text{if} \ X_n = 1 \\ 1-1/n & \quad \text{if} \ X_n = 0 \\ 0 & ...
2
votes
2answers
54 views

Modeling with Markov Chains and one-step analysis

I have set up the following model: Let $X_n$ be the number of heads in the $n$-th toss and $P(X_0=0)=1$. I can calculate the transition matrix $P$. Define $$ T=\min\{n\geq 0\mid X_n=5\}. $$ Then ...
2
votes
1answer
41 views

What is the intuitive difference between almost sure convergence and convergence in probability? [duplicate]

It is a standard fact in probability that almost sure convergence is stronger than convergence in probability. I can only see the differences in the proof. However, is there a way to view it ...
0
votes
1answer
20 views

How can I compute the mean of a sequence of random variables?

Suppose that I have a sequence of random variables where $X_1, X_2, \ldots$ where the pdf of $X_n$ is equal to: $$ f_n(x) = \begin{cases} (n-1)/2 & \quad \ -1/n < x < 1/n \\ ...
0
votes
1answer
36 views

Continuous Probability - Bus Arriving

I am trying to do the following question: Number 24 and number 42 buses arrive independently at the corner of Mayeld Road at a random rate of 3 and 4 per hour respectively. You arrive at the ...
3
votes
1answer
18 views

Moment generating function and convergent random variables

denote by $X$ and $X_n$, $n\in \mathbb{N}$, random variables and $r\in\mathbb{R}_+$ with $E=\mathbb{E}\left[ e^{rX} \right] < \infty$ and $E_n=\mathbb{E}\left[ e^{rX_n} \right] < \infty$ for all ...
0
votes
0answers
24 views

Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...
0
votes
0answers
26 views

Probability to get from A to C.

There has been a snowstorm and Bob is trying to drive from A to C. p and q are the probabilities that the two roads are passable. What is the probability that Bob can get from A to C? Note that ...
1
vote
0answers
39 views

Single, 6-sided die probability

I'm working on an assignment and I'm more or less new to stats. It might be the wording of the questions that's getting me as well. It deals with a regular 6-sided die. 1.a) What is the mean number ...
0
votes
0answers
22 views

poisson process(exercice)

Let $N_t$ a Poisson process and $S_{N_t}=X_1+...+X_{N_t}$. Let $A_t=t-S_{N_t}$ and $B_t=S_{N_t}-t$ 1)Show $P(B_t \geq x \ and\ A_t \geq y)=\frac{1}{E(X_1)} \int_{x+y}^{\infty} P(X_1 \geq u)du$ with ...