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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0
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0answers
5 views

Which multivariate Gaussian has the highest expected norm?

Let $X$ be an $n$-dimensional Gaussian vector with zero mean and covariance matrix $K$ given by: $$K_{ij} = \begin{cases} p_i(1-p_i) & i=j \\ -p_ip_j & i\neq j\end{cases}~,$$ where ...
0
votes
0answers
10 views

Probability problem possibly based on principle of inclusion exclusion

The problem reads as follows: Probabilities that Rajesh passes in Physics, Math and Chemistry are $p$, $m$ and $c$ respectively. Of these subjects, Rajesh has $75%$ chance of passing in at least ...
0
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0answers
12 views

Probability that One Can Find a Multiple of Uniformly Random Value

We define finite field $\mathbb{F}_q$, where $q=2p+1$ and $p$ is a large prime number (e.g. 256-bit). I pick uniformly at random value $\beta$ from the subfield such that $\beta>\frac{q}{2}$. ...
0
votes
0answers
14 views

Expected number of button clicks

Suppose we have $N$ buttons and each button can be clicked with probability $p_i$. The game stops when the player clicks the button with $i = 1$. What is the expected number of clicks? I am not able ...
0
votes
0answers
6 views

Confidence bands on a QQ plot

My task is to manually construct a QQ-plot with pointwise confidence intervals. If I understand everything correctly, this short manual uses intervals $$\Phi^{-1}(F_n(x)-k)\leq x \leq ...
0
votes
0answers
7 views

Replace probability mass function with continuous argument for asymptotic: how to reinterpret

I'd like some help reinterpreting the asymptotic result on a pmf. The asymptotic result is a continuous function, whereas the mass function is discrete. Say $P(Y(t)=k)=f(k,t)$. It is the case that ...
2
votes
2answers
68 views

Proving that $P(X=Y) = 0$ for any two continous random variables

I have the following question to prepare for a lecture at Uni but I've been stuck on this for a long time: Question: Let $Z$ and $V$ be independent with distribution $U[0,1]$. Show that ...
1
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0answers
12 views

Conditional Probability Distribution Notation

I've been reading a text containing an introduction to probability theory, and I ran into the following formula for conditional probability distributions. Note that $Pr(x)$ here is not the probability ...
-1
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4answers
39 views

If $P(A|B\cap C) = P(A|C)$ then $P(A\cap B|C) = P(A|C)P(B|C)$ [on hold]

Can anybody prove $P(A\cap B|C) = P(A|C)P(B|C)$? I got it up to here. $P(A\cap B|C)= P((A\cap B)\cap C) /P(C)$. Can somebody continue it please or start it from the beginning? Update: It is also ...
6
votes
1answer
152 views

Why weak law of large number still alive?

I know the difference between WLLN and SLLN in terms of a convergence type. Then, as revealed in any statistical textbook saying sufficient conditions to two theorems are the same, I think that we do ...
1
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0answers
20 views

Difference between P(A,B) vs P(A∩C) vs P(A.B) vs P(AB)

P(A,B) P(A∩B) P(A.B) P(AB) Above 4 statements looks almost similar to me. Can anybody define if there is any difference between above 4 and compare them in detail.
0
votes
1answer
28 views

Is $P( (A\cap B)\cap C)$ equal to $P(A\cap C) P(B\cap C)$?

Is $P( (A\cap B)\cap C)$ equal to $P(A\cap C) P(B\cap C)$? In a proof I doubtfully used this equation. Is it correct? But I am not sure about it. Can somebody confirm its validity? If possible, can ...
1
vote
0answers
12 views

Conditional Probability

There are two local factories that produce radios. Each radio produced at factory $A$ is defective with probability $.05$, whereas each one produced at factory $B$ is defective with probability $.01$. ...
0
votes
1answer
16 views

Combinations and paritioning

Nine workers are assigned to nine jobs. Two of the jobs are considered bad, four are considered average, and three of the jobs are considered good. The 9 workers consist of seven men and two women. If ...
0
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0answers
40 views

Some questions about computing integrals?

I am trying to compute some integrals in the paper. My question is about the example on page 119 after Lemma 5.2.8 on page 118. Why $$\int_0^T e^{-\alpha_2(\pi)} = \int_0^T ds ...
2
votes
1answer
28 views

Probability that a set of $n(n+1)/2$ elements will contain $1… n$ elements, respectively, of $n$ possibilities

We opened a 'fun size' bag of Skittles this afternoon, and it contained 5 yellow, 4 red, 3 blue, 2 green, and 1 purple Skittle. If the Skittles only come in these 5 colors, they are chosen randomly ...
-1
votes
0answers
19 views

Conditional Probability on a Mixed RV

TLDR: How do you show $P(Y\in\Omega|X=x_0)=1$ when $Y$ is a mixed RV. Let $(X,Y)\to\mathbb{R}^{dx+dy}$ be a random vector distributed according to a density $f_{X,Y} (x,y)$. Let $X$ be a continuous ...
1
vote
1answer
21 views

Two uncorrelated random variables both taking only two values are independent

Let X and Y be random variables both taking only two values. Show that if they are uncorrelated then they are independent.
0
votes
0answers
8 views

How to express $E_{\theta}(S|T)$ in terms of $T$ (complete, sufficient, unbiased)only?

Suppose that a statistic $S$ is unbiased for a parameter $\theta$, while $T$ is complete, sufficient, and unbiased for $\theta$. express $E_{\theta}(S|T)$ in terms of $T$. Can some one give me a hint? ...
5
votes
1answer
28 views

Find $\sum_{k=0}^{\infty}(1-1/n)^{2k}\frac{e^{-n\theta}(n\theta)^{k}}{k!}$ (the variance of $(1-1/n)^{X_1+\cdots+X_n}$)

Given a random sample $X_1,\ldots,X_n$ from Poisson distribution with an unknown parameter $\theta>0$.$T:=(1-1/n)^{X_1+\cdots+X_n}$. Find $\operatorname{var}(T)$. My work: I find $T$ is a UMVUE ...
2
votes
1answer
36 views

Proof that absolute value of a random variable is a random variable

Is this proof correct?: Proof: Suppose that $X$ is a random variable on a probability space $\{\Omega, \mathcal{F}, \mathbb{P}\}$. Suppose $x \in \mathbb{R}$ and $x \geq 0$. Then $\{|X| \leq x\} = ...
2
votes
1answer
21 views

$X,Y \sim $iid $\operatorname{Exp}(\lambda),\ P(X \le t, X + Y > t)$?

$X$ and $Y$ are 2 identical exponentially distributed independent random variables. $X,Y \sim$ iid $\operatorname{Exp}(\lambda)$ What's the following probability? $$P(X \le t, X + Y > t)$$ I ...
1
vote
1answer
23 views

Conditional expectation of iid nonnegative random variables

I am studying Ross's book, stochastic processes. There is the following lemma: Let $Y_1, Y_2, ... , Y_n$ be iid nonnegative random variable. Then, $E[Y_1+ \cdots +Y_k | Y_1+\cdots+Y_n=y] = ...
-1
votes
1answer
49 views

Expectation of a symmetric function about zero [on hold]

I understand that the median of a symmetric function about zero is zero because 50% of the mass is contained either side of it. It is easy to prove this result for the Expectation,however I do not ...
2
votes
1answer
23 views

Probabilistic Conditioning. Please help me finish the solution to the problem.

I have to solve the following problem: The real random variables $X$ and $Y$ are independent and have a uniform distribution $U([0,1])$. Find $$\mathbb{E}\left( \frac{3 X-Y+1}{\sqrt{X+Y+1}} | \quad ...
1
vote
1answer
22 views

Bivariate normal distribution of points

I would like to generate points (x,y) in a 2-D plane that has a circular normal distribution similar to this: I found multiple terms for describing a "circular normal distribution" and yet, I'm not ...
1
vote
1answer
26 views

How can we measure the accuracy of prediction algorithm?

We have created a prediction algorithm, which predicts the chances of confirmation of ticket based on some parameters, and gives the prediction in percent. Now, how do I measure how close the ...
0
votes
1answer
19 views

What is the probability of obtaining the right, left, Ace, King, and Queen of trump in a 5-card hand in euchre? [on hold]

Take into account that in order for that hand to be the best hand, the dealer must also flip over a card that matches the suit that your best hand is for. (If you have the J of spades, Jack of clubs, ...
-1
votes
1answer
21 views

How many different hands are possible if two spades are drawn from a deck of cards? [on hold]

Two cards are drawn from a deck of 52 cards. If both cards are spades, how many different hands are possible? (Note: A hand of cards is the collection of cards a player is holding, a standard deck ...
1
vote
0answers
44 views

Number of rewards before death

I have a question regarding Poisson events with death. Assume time is continuous $t\in[0,\infty)$. A person may die with intensity $\delta$. Conditional on being alive, he may achieve a reward with ...
1
vote
1answer
20 views

How do i calculate the probability of the relay in the circuits?

I am trying to solve my following probability question but i can't see how to make any progress. Any help will be highly appreciated Question: The probability of the closing of the i-th relay in the ...
1
vote
5answers
38 views

Probability - consecutive numbers

Question: Three numbers are selected out of the first 30 natural numbers. What is the probability that none of them are consecutive? I know that the total possibilities will be $^{30}C_3$ ...
8
votes
1answer
49 views

Expected value and variance of ratio of two sums of two sets of random variables

Let $X_1,X_2,\ldots,X_n$ be iid $\operatorname{Gamma}(\alpha,\beta)$ random variables. Suppose that, conditionally on $X_1,X_2,\ldots,X_n$, the random variables $Y_1,Y_2,\ldots,Y_n$ are independent ...
-3
votes
0answers
27 views

a question about sample space [on hold]

How to represent the following statement mathematically: "The event $\{A_n \text{occurs infinitely often} \}$ is $\{ \omega \in \Omega | \omega \in A_n \ \text{for infinitely many values of} \ n ...
0
votes
1answer
26 views

N balls having M different colors in a box, how many times do I need to pick to get one particular color?

There are $N$ balls of $M$ different colors in a box i.e $c_1$ balls of color $1$ and so on. $c_1 + c_2 + \dots + c_M=N$, $c_1, c_2, \dots, c_M$ are known. We are looking for a ball of a particular ...
1
vote
0answers
11 views

Estimating the expectation of a derivative [on hold]

Assume $Y$ is a continuously differential function of $X$. Given i.i.d. data $(x_i,y_i)_{i=1}^n$, I would like to estimate $E\left[\left.\frac{\partial Y}{\partial X}\right|_{X=X_0}\right]$. ...
0
votes
0answers
26 views

Identifying and separating two different distributions in a set of mixed data

Data The data at hand comprises distances between successive points of known location, which occur with set limits (red line is of finite, known length): Points are chosen in succession, one after ...
0
votes
1answer
39 views

poisson distribution probability problem

I am working on a Poisson distribution problem stated in the main question and got stuck and do not know how to proceed as I did not understand the next question on how to work it out The following ...
0
votes
1answer
28 views

If $w$ is a discrete random variable then is $P(w|x)$ a density or mass function? [on hold]

$w$ is a discrete random variable. $x$ is a continuous random variable. Then should I denote $P(w|x)$ as a probability density function or probability mass function, and why ?
0
votes
0answers
25 views

Probability and Expected number of games played

I am wondering how I would I apply Markov chains or martingales to solve problems of the following type: Example : Two players play games against each other until either of them wins 3 games in a ...
-1
votes
1answer
18 views

Probabilities for selections from a set

This seems like it would be a common question, but I can't come up with a search that provides an answer to my question, so I'm asking it here. I have a set of unique numbers ...
6
votes
3answers
54 views

distribution of one random over the sum of random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables with $X_i\sim Gamma(\alpha_i,\beta)$. Define $U_i=\frac{X_i}{X_1+\cdots+X_n}$ for $i=1,2,\ldots,n$. Show that $U_i\sim ...
0
votes
0answers
32 views

The Gambler's Ruin without using random walks

This is more of a doubt. I understand that this problem can be described with Markov chains and the recursion solved without much trouble. However I've seen that some people casually say that $$ ...
2
votes
1answer
32 views

How to compute the expected value of one random variable over sum of iid random variable

If $X_1,\ldots,X_n$ are independent identically distributed positive random variables, prove that $E(\frac{X_i}{X_1+\cdots+X_n})=\frac{1}{n}$, $i=1,\ldots,n$. Can someone give me a hint?
-2
votes
1answer
26 views

Probable winner of last coin game of a series, where winner from one game has disadvantage the next game?

Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner is the first person to obtain a head. They play this game several times, with the stipulation that the loser ...
1
vote
2answers
41 views

Random walk in one dimension with different probabilities

As the title suggests, I'm concerned with a typical random walk problem, where the probability to go right is $p$ and the probability to go left is $q=1-p$. I was trying to find the probability of ...
1
vote
2answers
50 views

$X$ and $Y$ are independent Poisson$(\lambda)$, $\lambda\sim\mathrm{exp}(\theta)$. What is the conditional distribution for $X$ given that $X+Y=n$?

To clarify, the parameter $\lambda$ is a random variable with exponential distribution and parameter $\theta$. Can someone please tell me if I've correctly computed the conditional distribution for ...
-5
votes
1answer
33 views

Expected Value and Expected Variance Probability [on hold]

Suppose a production line manufactures ball-bearings with a radius that is uniformly distributed between 1.8mm and 2.2mm. What is the probability of (a) the expected value of the volume, and (b) the ...
0
votes
1answer
27 views

probability distribution of the winning amount

Be A_n the event that a worker spends to process certain component with probabilities according to the table below: For each piece processed, the worker earns a fixed US 2.00, but if he processes ...
-1
votes
0answers
24 views

Probability, normal distribution, car collision [on hold]

There is a question in the book Principles of Statistics by M.G. Bulmer which I'm stuck on. Here goes: The reaction time of two motorists A and B are such that their braking distances from 30 m.p.h. ...