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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1
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2answers
31 views

Why $p\{N>n\}=p\{X_1+…+X_n\leq x\}$.

Let $(X_k)$ a sequence iid of random variable uniform on $[0,1]$. Let $x\in]0,1[$ and $N=\min\{n\geq 1\mid X_1+...+X_n>x\}$. Why $$p\{N>n\}=p\{X_1+...+X_n\leq x\} \ \ ?$$
0
votes
1answer
16 views

Let X have density 2t on 0 < t < 1 and Y be uniform on the interval (0, 10) and independent of X. Find the density of Y/X.

Let X have density 2t on 0 < t < 1 and Y be uniform on the interval (0, 10) and independent of X. Find the density of Y/X. I have no ideas how to solve it
2
votes
1answer
28 views

Coin tossing: Streak count

I have a special request with regards to probability. Let's say I toss a coin 400 times. What I need to know is the average number of streaks of various lengths within such a sample. How many ...
0
votes
0answers
8 views

bias reduction when the bias depends on the true parameter

Let's say we estimate a parameter, $\theta$, by $\hat{\theta}$. For this estimator we have the following property that $$\hat{\theta}\to_{p}\theta+f(\theta)$$ where $\to_{p}$ denotes convergence in ...
-4
votes
1answer
29 views

probability question of balls [on hold]

what is the chance of getting at least one defective item if 3 items are drawn randomly from a lot containing 6 items of which 2 are defective?
2
votes
0answers
20 views

normal squared characteristic function derivation

I'm trying to derive the normal squared characteristic function, there's already a question on this but the answer has a part which is "proved as an excercise" which I try to do here. Is my proof ...
2
votes
3answers
141 views

What is the probability that 5 randomly chosen cards in a deck add up to 40 or greater?

I have made a probability game, where you have to pull out any 5 cards without looking (from a deck of 52 cards), and if all five cards add up to 40 or more, they player pulling the 5 cards from the ...
0
votes
0answers
21 views

Sum of two independent Continuous-Time Markov Chains [on hold]

This is the first time I have come across a question involving the sum of two independent continuous time Markov Chains.I know you can find the sum of two random variables Z = X + Y by finding the ...
0
votes
0answers
11 views

If two sequence of random variables $X_n$ and $Y_n$ converge in Probability to X and Y, then X = Y a.s

If two sequence of random variables $X_n$ and $Y_n$ converge in Probability to $X$ and $Y$, then $X = Y$ a.s. Idea: I want that $P(|X-Y|> \epsilon) = 0$, for every $\epsilon >0$. We can ...
6
votes
2answers
125 views

Rolling two dice, what is the probability that two consecutive $7$s happens earlier than a $12$?

Alice and Bob are playing a game involving two dice. If a sum of 12 appears, Alice wins and they stop playing. If a 7 appears twice in a row, Bob wins and they stop playing. What is the probability ...
-1
votes
1answer
18 views

In how many different ways can a person vote based on the problem below? [on hold]

There are 9 people on the ballot for regional judges. Voters can vote for any 4. Voters can choose to vote for 0, 1, 2, 3, or 4 judges. In how many different ways can a person vote?
3
votes
0answers
28 views

Why is this class closed under difference?

We have two independent random variables $X\perp Y$ involving three spaces: $(\Omega,\mathcal{A},P), (E,\mathcal{E}), (F,\mathcal{F}).$: $$X:\Omega \rightarrow E,\ Y:\Omega\rightarrow F$$ My book says ...
0
votes
0answers
21 views

How to compute P(|X - E_Y[h(y)]| < c)? [on hold]

Consider the discrete random variable $Y$, the continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, ...
0
votes
3answers
23 views

Events with States

Bob and I are playing a game with an unfair coin that is rigged to come up heads with probability $\frac35$ and tails with probability $\frac25$. Bob goes first, we take turns, and the first player ...
1
vote
0answers
24 views

vase with blue and red balls

At first I hope this is not a duplicate post. I tried to find it but I have not found it. I hope that someone could help me with understanding the exercise. This question is about a vase with r red ...
0
votes
1answer
39 views

Question about Measure Theory [on hold]

Let $(\Omega, U, P)$ be a measure space and X be random variable and its distribution function $F_x(x)=P(\{\omega: X(\omega)\le x\})=P(-\infty , x]$ and the function F is continuous at x. If the ...
1
vote
1answer
27 views

interpreting wording of probability question

Two dice are rolled, and the sum of the face values is six. What is the probability that at least one ofnthe dice came up a three? I want to make sure that I am interpreting the language right when ...
2
votes
2answers
38 views

Martingale definition

To prove that one process is Martingale, generally we prove 3 things : 1) X is adapted. 2)$$ \mathbf{E} ( \vert X_n \vert )< \infty $$ 3) $$\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n $$ I ...
1
vote
1answer
17 views

What is the relation between$ P(A|B)$ and $P(A|B')$ for both independent and not independent events?

Let $A$ and $B$ be two events. If they are independent, how are $P(A|B)$ and $P(A'|B)$ related, if at all? If they are not independent, how are $P(A|B)$ and $P(A'|B)$ related, if at all? I've noticed ...
3
votes
0answers
18 views

Seven-Card Stud with Random Hand Selection

I was recently confronted with a number—$2727707$, actually—that started a short train of thought while I was placed on hold. (This seems to happen quite often: both the observation of unusual ...
3
votes
2answers
32 views

Conditional probability of exponential random variable

This question comes directly from a chapter in Gut's "Intermediate Probability" that focuses on conditional probability. I'm using this problem as more practice solving conditional probability ...
5
votes
3answers
70 views

Suppose a city with Three type of coins ?!

in a city we have tree type 1 dollar, 2 dollar, 3 dollar of coins. we want to pay for a 20 dollar product. how many ways we can pay for a 20 dollar product, if the seller has no money and number of 1 ...
-1
votes
1answer
34 views

Marginal distribution of uniform distribution conditioned on poission?? [on hold]

Let $N$ be a Poisson distribution with parameter $\lambda$. Conditioned on $N$, let $X$ be uniformly distributed. What is the marginal distribution of $X$? ( this is one of my final exam problems ...
0
votes
1answer
28 views

Geometric Random Variables

I have a question that involves a certain criteria of a random variable as shown: The random variable $X$ has the distribution $Geo(0.2)$ and I would love it if someone could explain what the ...
0
votes
0answers
54 views

What is the probability that 5 randomly chosen cards in a deck add up to 40 or more? [on hold]

I have made a probability game, where you have to pull out any 5 cards without looking (from a deck of 52 cards), and if all five cards add up to 40 or more, they player pulling the 5 cards from the ...
1
vote
0answers
50 views

proving a statement on Measure theory [on hold]

Consider $(\Omega, U, \mu)$ be a measure space and X be an integrable function and for $A$, $\{A_n\}\in \mathscr{U};n\in \Bbb N$ I need to show that $\int_{A_n}X d\mu \to_{n\to \infty}\int_A Xd\mu$ ...
2
votes
1answer
55 views

What's the probability that the first four children born are boys and the last two children born are girls?

I'm having some problems with determining how to calculate a question about the gender proportion in newborns in some random family. A family consists of 6 children. The probability of a boy being ...
0
votes
1answer
13 views

Variable drawn from a normal distribution

What exactly is the meaning of a "variable drawn from a normal distribution"? I know what a normal distribution is, but my main exposure to "variables" is from calculus, so I have a hard time ...
1
vote
2answers
31 views

Odd Power terms of binomial theorem proof

I want to acquire all the terms of $(p+q)^n$ where the power of p is odd. Note that $p=1-q$ ($p$,$q$ probabilities) Ex. For $(p+q)^2=p^2+q^2+2pq$ I want to acquire only $2pq$(only term with odd ...
0
votes
0answers
22 views

Combining Sample Means and Variances

Let us assume we have several samples of unknown size but known mean value $\mu_{i}(x)$ and known variance $\sigma_{i}^{2}$. Now we want to calculate the mean value and variance for the total ...
3
votes
3answers
30 views

Symmetry in Probability Around a Particular Phenomenon in Time?

This has been hurting my brain substantially, recently. I'm not sure if I'm failing to make connections or if I see connections but am weary of their relevance. In my text the author claims that ...
0
votes
0answers
22 views

Question concerning invariant distribution

Let us consider the Markov chain $(X_n)_{n \in \mathbb{N}}$ with state space $I = \{0,1\}^m$ and transition probabilities $$ p_{xy} = \begin{cases} m^{-1} &\mbox{if } \vert x - y \vert = 1 \\ 0 ...
2
votes
2answers
49 views

Confidence Interval - Cigarette HW Question

Due to a lack of general student discussion on the message board my class uses, I've decided to ask this here. I want to know if I proceeded with this question correctly and if my choices were ...
-1
votes
0answers
41 views

Interesting and challenging problem [on hold]

I've been given this problem to solve, but didn't succeed until now. Can you help me? A city has 5 billion paper money (bills) in circulation. Thirty million of them are taken daily to the bank ...
1
vote
3answers
23 views

Probability of choosing two bulbs with the same rating given that one has a specific rating

I am trying to teach myself statistics, and working through Jay DeVore's excellent text of "Probability and Statistics for Engineering and the Sciences". The problem is as follows: We have box of the ...
0
votes
1answer
18 views

Independence - Probability and Statistics

Any help on this problem is greatly appreciated! I'm completely stuck School board officials are debating whether to require all high school seniors to take a proficiency exam before graduating. A ...
0
votes
0answers
9 views

probability generating function moments for the multivariate case

Suppose ${\bf X} = (X_1, \ldots, X_d)$ is a non-negative integer-valued random vector, with pmf $p$, I tried to extend the results of the univariate generating function to the multivariate case, is ...
0
votes
2answers
36 views

proof of conditional probabilities

show that if the conditional probabilities exist then $$p(A_1\cap A_2 \cap \cdots \cap A_n) = p(A_1)p(A_2\mid A_1)p(A_3\mid A_1\cap A_2)\cdots p(A_n\mid A_1\cap A_2 \cap A_3\cap\cdots\cap A_{n-1})$$ ...
1
vote
1answer
21 views

How to make 4608 combinations with these choices? (Probability, permutations/combinations)

This problem has been giving me a lot of trouble... Freeze King claims to offer 4,608 different ice cream cups. A customer can choose from 3 sizes, 4 flavors; a waffle cone, sugar cone, or cup; ...
0
votes
0answers
5 views

p-average compound metric

I'm trying to prove that probability space metric defined as $d(X,Y)=(\mathbb{E}|X-Y|^p)^{1/p}$ is a metric indeed. Literature states that $d(X,Y)=0$ implies $Pr(X=Y)=1$, but no further explanations ...
1
vote
1answer
29 views

probability of 26 letters

A monkey at a typewriter types each if the 26 letters of the alphabet exactly once, the order being random. A. What is the probanility that the word HAMLET appears somewhere in the string if letters? ...
0
votes
1answer
24 views

positiv Martingale process

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2ds}$$ is a martingale which is positiv and has a mean=1 $$\theta is continuous ...
0
votes
1answer
50 views

A measure theory question-1 [on hold]

Let $ (\Omega, \mathcal U, P)$ be a measure space and any events $A_1, A_2, A_3 \in \mathcal{U}$ And $ B$ is defined as event of occurrence of at least one of these three events. First I need to ...
0
votes
1answer
48 views

Probability of last cheese

I hope that someone could help me with understanding the exercise. In a cycle shaped house there are n chambers. In this house there is a mouse and each chamber has cheese except the room where the ...
0
votes
2answers
39 views

A probability theory question [on hold]

let X be a rondom variable and coonsider a non-negative function $g: \Bbb R \to \Bbb R^+$ Please help me sshowing this following statement; for $r\in \Bbb R^+ $, $$P(g(X)\gt r) ...
1
vote
3answers
52 views

Intuition behind independence result

The following problem is from Wasserman's $\textit{All Of Statistic}s$. I have worked through the algebra to arrive at the result, but it still seems very strange to me, so I would appreciate any ...
0
votes
0answers
10 views

Cardinality of maximum independent set for a given degree distribution

Consider undirected graph $G(V,E)$. Assume that $f_n(k)$ be the probability mass function of degree of a vertex in $G$. Further, assume that $f_n(k)$ is an strictly decreasing function of $k$ with ...
2
votes
1answer
32 views

Expected value - product of functions of uniformly distributed variables

We have $n$ random variables $X_1,...,X_n$, $n\geq 2$, where $X_i∼U(0,1)$ and all of them are iid. Let $ Z=\min(X_1,...,X_n)$ and $ \overline{X} = \frac{1}{n}\sum_{i=1}^{n}{X_i}$. Calculate ...
-1
votes
1answer
32 views

Lemme itô and Martingale [on hold]

I want to to find values of $a$, $b$ such that the process: $$e^{W_{t}^2+at+b\int_\limits{0}^{t}W_{s}^2\,ds}$$ be a martingale Could you please help me do that Thank you
6
votes
2answers
40 views

simulating a fair six with a four equal sector spinner

Whist teaching basic probability I needed a group to use a fair four sector spinner but I'd none left. I gave them a die asking them to disregard 5,6 should they arise. The problem got me thinking ...