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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5
votes
1answer
22 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
28 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
18 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
20 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
45 views

Expectation of a symmetric function about zero

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
20 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
16 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
17 views

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

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
19 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 ...
0
votes
0answers
33 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
3answers
25 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
47 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
25 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
25 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
37 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 ...
-1
votes
1answer
26 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
53 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
40 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
25 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. ...
-1
votes
1answer
37 views

What is a nice, clean proof to show that a fair coin toss satisfies axioms of probability?

If we assume H=Heads T=Tails and we're dealing with a fair coin what is a good way we can show that Kolmogorov Axiom has been satisfied?
-1
votes
0answers
15 views

Expected Value of covariance [on hold]

You have an urn with balls that are either red or blue, and striped or not. What is the probability distribution that describes the number of blue balls drawn given the number of striped ones drawn? ...
0
votes
2answers
31 views

Having an independent event with animals

In a building for 24 apartments. It is known that there is only one dog in 8 apartments and a single cat in 6 apartments. How many apartments must have cat and dog for events "have dog" and " have ...
4
votes
2answers
63 views

Confusing probability problems based on product rule and combinations

I am going thru probability exercise. Faced first problem: Book Q1. Ten tickets are numbered 1,2,3,...,10. Six tickets are selected at random one at a time with replacement. What is the ...
1
vote
0answers
32 views

Normal and poissonian probability problems

I am working on a problem with a normal probability distribution but I am unsure of the results I calculated the probability asked for but still hesitate regarding the output and especially the first ...
1
vote
1answer
50 views

Probability of no 6 or no 5 when dice is rolled n times

Can anyone guide me in the general direction of the answer to the following: A die is rolled $n$ times $$A = \text{no $6$s}$$ $$B = \text{no $5$s}$$ $$P(A\cup B) = \;?$$ I am first finding $P(A)$ ...
0
votes
2answers
44 views

Independence between conditional expectations

Suppose $(\Omega, F, P)$ is a sample space, $X$ and $Y$ random variables, and $N$ and $M$ sub sigma algebras of $F$. I know that $E(X\mid N)$ and $E(X\mid\{\emptyset, \Omega\})$ are independent. ...
1
vote
1answer
51 views

The ant is moving through the coordinate system, Started at $(0,0)$ to $(4,4)$. What is the probability that the ant will find food at $(3,2)$?

The path to the $(3,2)$ is $3+2 \choose 3$ or $3+2 \choose 2$. Total path is $4+4 \choose 4$ And the probability is : $ \frac{3+2 \choose 3}{4+4 \choose 4}$ = $ \frac{5 \choose 3}{8 \choose 4}$ = ...
-1
votes
1answer
17 views

Length of random interval in $[0,1]$ that contains point $x$ [on hold]

The interval $(0,1)$ is divided by a point uniformly distributed in the interval $(0,1)$. Given $x \in (0,1)$, find the average length subrange which contains the point x. Show that this average is ...
0
votes
2answers
24 views

How to prove expectation exists (or improper integral converges)

How can I prove this improper integral converges, or give a counterexample? $$\int_{-\infty}^{\infty}x^n p(x)dx$$ where the only thing we know about $p(x)$ is $$\int_{-\infty}^{\infty}p(x)dx = 1 $$ ...
-1
votes
1answer
22 views

Are the converses of the following special cases of conditional expectation also true?

Let $X$ be a random variable, and $N$ be a sub sigma algebra of the underlyign sigma algebra of the sample space. if $X$ is in $L^1$ and measurable wrt $N$, then $E(X|N)=X$ a.e.. Is it true that ...
1
vote
1answer
23 views

Justification for Interchange of integral and sum

Let $\mu$ be a probability measure and $t\in\mathbb{R}$. I would like to write this equality $$\int_{\mathbb{R}}e^{ixt}d\mu(x)=\sum_{n\geq0}\frac{(it)^{n}}{n!}\int_{\mathbb{R}}x^{n}d\mu(x).$$ This is ...
2
votes
2answers
29 views

A conjecture about generating algebras on a probability space

Suppose that $(X,\mathscr F,\mathbb P)$ is a probability space. Let $\mathscr E\subseteq\mathscr F$ be an algebra (i.e., it is a non-empty collection closed under complementation and finite unions) ...
0
votes
4answers
24 views

Probability of extracting twice same ticket out of 4 pcs

I have just extracted from 2 consecutive tries the same ticket out of 4. How do I calculate the probability of such an event?
1
vote
2answers
37 views

Taking a ball from an urn after passing a randomly chosen ball from another urm

An urn contains four blue balls and three white balls. A second urn contains five blue and four white balls. Pass up a ball from the first to the second urn and then extracted a ball second urn. I ...
0
votes
0answers
18 views

probability of mutations occuring by chance mutually exclusively in cancer

I have a dataset that tells me if there are mutations in any of 500 genes in 100 cancer patients. Some patients have 0 mutations and some have >200. Genes generally work in networks, some of the genes ...
0
votes
0answers
58 views

From the binomial distribution

A single cell can either die, with probability $0.1$, or split into two cells, with probability $0.9$, producing a new generation of cells. Each cell in the new generation dies or splits into two ...
4
votes
2answers
50 views

Coin toss problem, get exactly 2 heads in 5 tosses

Suppose we toss a fair coin until we get exactly 2 heads. What is the probability that exactly 5 tosses are required? My try: We have to make sure that the first 4 tosses does not have 2 ...
2
votes
1answer
29 views

Birhdays: find the probabilities for the various configurations of the birthdays of 22 people

Let S,D,T,Q stand for simple,double,triple and quadruple, respectively: So, for example: the probabilities of 22 simple birthdays(22 person have birthdays in different days) are $ P(22S) = ...