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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2
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1answer
21 views

convergence in probability: speed of convergence

I am not sure if the title appropriately describes the question. I will appreaciate any ideas. Suppose $\{X_n:n\geq 1\}$ is a sequence of random variables defined on a common probability space. ...
0
votes
1answer
19 views

Sums of partially dependent Bernoulli random variables

I am looking for any kind of Chernoff type large deviation bound for the following random variable: $$X = \sum_{i=1}^NX_i$$ where each $X_i$ is an identically distributed Bernoulli random variable ...
0
votes
0answers
18 views

On the Chernoff bound

Recently, I saw the Chernoff bound written as follows. Let $X_1,X_2,\ldots,X_n$ be drawn i.i.d. on alphabet $\mathcal{X}$ and let $f:\mathcal{X}\to [0,1]$ be any function. Let $\mathbb{E}f(X_1) = ...
1
vote
0answers
9 views

Total variation distance and couplings

The total variation distance between two measures $\mu$ and $\nu$ can be shown to equal the infimum over all couplings $(X,Y)$ where $X\sim\mu, Y\sim\nu$ of $P(X\neq Y).$ What is the supremum of ...
0
votes
0answers
19 views

How many uniform samples are needed to hit every element

Let $D$ be the uniform distribution over $\{1,\ldots,n\}$. How many draws from $D$ (asymptotically) are needed such that with high probability (say $2/3$) all $n$ elements were drawn at least once?
1
vote
1answer
32 views

Circuit probability question regarding sum of a random number of independent random variables

Suppose we have n circuits that operate in a home. Each one will live according to an exponential random variable with rate λ. If X denotes the time at which a circuit first dies (i.e. the first circuit ...
0
votes
2answers
30 views

Get the distribution of $X|Y=y$ given this joint probability density function

Given the joint probability density function $f(x,y) = \lambda^2 \exp(-\lambda y)$ with $0 < x < y.$ How do I get the distribution of $X|Y=y$ ? Thanks in advance!
0
votes
1answer
27 views

Find probability of a Poisson Process

I have a Poisson process $N(t)$ with $\tau$ for customer arrival in a shop. $N(t)$ is spllitted with two types of arrival (male and female). It can be shown that the process is a combination of two ...
3
votes
0answers
49 views

Why is $F$ continuous?

Why is the function: $F: P(\mathbb R) \to \mathbb R$, $F(X) = \int_X e^{-x} dx$ a continuous function? How to prove such a thing? Does it even make sense to talk about the continuity of such a ...
-1
votes
2answers
22 views

Conditional day distribution probability

Let $X$ be a random day of the week, coded so that Monday is 1, Tuesday is 2, etc. (so $X$ takes values 1, 2,..., 7, with equal probabilities). Let $Y$ be the next day after $X$ (again represented as ...
2
votes
1answer
34 views

Random variable to the power of minus one?

I have a definition, it goes as follows: $\Pr$ is probability. $X$ is a random variable. $x\in\mathbb{R}$ $$Pr(X = x) = \Pr(\{ \omega\in\Omega \mid X(\omega)=x\})$$ So for example for a dice of 6 ...
1
vote
1answer
45 views

Coin and Lottery Question

Attempt: For the first part I presume I use bayes' theorem? For the second part, I can't count the number of ways of such a sequence. Thanks.
2
votes
0answers
12 views

Estimating the Average and Standard Deviation of a Population based on a Sample with Missing Data with Known Ranks

I need a way to shows me how the parameters of PDF, log-normal in this case, can be estimated based on a set with missing data points at the tail end of a sample. For example, Consider we had 20 ...
2
votes
2answers
49 views

Calculating probability for forming a triangle

I am having trouble coming up with a solution for this problem: There is a stick of unit length. We break it into two parts. Now, we pick the bigger one and break it into two parts. I want to ...
2
votes
3answers
34 views

Distribution of a fractional part of the sum of uniform RVs

I had a question in class not long ago which I couldn't solve. I've been digging into it for a few hours now but I can't find the right direction. So the question is: Let $ U_1,..,U_n$ be I.I.D ...
0
votes
0answers
13 views

Simulate ICA Source Signal

I am using the fastICA package in R for a matrix of time series information. However, if I wanted to simulate the process for risk management purposes how exactly could I do this? For example lets ...
0
votes
2answers
27 views

The random variable $ Z = 1-F(X)$

I will formulate the theorem (with no proof) if $X \in \mathbb{R}$ is a random variable with continuous distribution function $F$ then the random variable $Z = 1-F(X)$ has a uniform distribution on ...
-1
votes
2answers
28 views

Does the order in a circular arrangement matter?

I posted a question a while ago: Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs. My question here is: imagine a ...
3
votes
1answer
55 views

Play until one is bust

If player A and B have $a$ and $b$ millions pounds respectively, where $a,b$ are natural numbers. They play a series of games in which the winner receives one million pounds from the loser (draws ...
1
vote
1answer
26 views

Custom Weighted Formula

I'm in need of a mathematical formula that will be ultimately utilized in any programming language that would give me a value that I could ultimately sort or rank by. I have 2 variables: variable 1 = ...
2
votes
3answers
369 views

Stars and Bars vs PIE

I randomly made up this question so I could check: There are $3$ kids and $6$ gifts, how many ways to distribute so that each kid has at least one gift. Obviously, $**|**|**$ there are ...
3
votes
1answer
28 views

Poisson Process Derivation.

I was looking at a derivation for the poisson process , which tells the number of events occurring in time $t$ , I came across the following differential equation : $\frac{d}{dt}(P_n(t))$ = ...
-3
votes
0answers
14 views

Combining experts probability judgements [on hold]

I seek for function that combine list of probabilities of event into one probability value. Each probability is given by expert. This function should fulfill: $Pc(\{1.0\}) < Pc(\{1.0,1.0\})$ ...
0
votes
1answer
38 views

Find the CDF of a function of two random variables

The joint probability density function of two continuous random variables $X$ and $Y$ is: $$f(x,y) = \begin{cases} 6x,& 0\leqslant x\leqslant y,\ 0\leqslant y\leqslant 1\\ 0,& \text{ ...
2
votes
1answer
24 views

Degree of Polynomial in Centered Moments of Gamma$(n,1)$

I'm interested in the degree of the polynomial in $n$ of the expression for the $k$-th central moment $$ E((X_n - n)^k) $$ where $X_n$ is a Gamma$(n,1)$ random variable, that is, the sum of $n$ ...
1
vote
0answers
12 views

conditional density function with respect to a statistic.

I have read a formula in a textbook: $$f_{Y|S}(y|s;\theta)=\frac{f_Y(y;\theta)}{f_S(s;\theta)}$$ $Y$ is a random vector and $S$ is a statistic of $Y$. According to the definition of conditional ...
4
votes
1answer
24 views

Largest jumps of a spectrally positive $\alpha$-stable process

Let $X(.)$ be a (strictly) $\alpha$-stable process (with $\alpha \in (1,2)$). Assume also that $X(.)$ is spectrally positive (its Lévy measure is concentrated in $[0,+\infty)$). I am looking for a ...
-1
votes
0answers
26 views

different combinations of numbers [on hold]

Can anyone help me pick 1,000 combinations of six numbers from two separate pools of numbers - five different numbers from 1 to 75 and one number from 1 to 15?
-3
votes
1answer
21 views

what is the expected number of times a 6 will be rolled? If a 6 is rolled 10 times what can be said about the die? [on hold]

A die is rolled 20 times what is the expected number of times a 6 will be rolled? If a 6 is rolled 10 times what can be said about the die?
0
votes
1answer
11 views

Hitting times of a biased continuous time random walk

Let $X_{s \geq 0}$ be a continuous time random walk on $\mathbb{Z}$, i.e. waiting times between jumps are exponentially distributed with mean one. The random walk is biased: $\mathbb{P}(X_s\text{ ...
0
votes
0answers
18 views

Finding upper critical value with Chebyshev's inequality

Consider $X$ is a Poisson random variable with distribution $X$~$Pois(\theta)$. I define the mean in my hypothesis as $\lambda$ and nominal significance level $\alpha$. Null hypothesis $H_0 : ...
-1
votes
1answer
49 views

Cards probability problem

Two players; the dealer and a player. The player is given three cards face down. The dealer turns over a 2 (let's say of hearts). Before the player turns any cards over, what is the probability that ...
2
votes
1answer
26 views

Symmetric function of two normal distribution implies bilinear

This question is related to my previous question which was partially answered my @MichaelHardy. Let $X$ and $Y$ be two independent standard normal random variables. Now, suppose that ...
0
votes
1answer
21 views

Question involving the PDF of a function of a random variable.

I'm trying to understand the setup for problem 3.1, from M.G. Bulmer's Principles of Statistics (Dover, 1967). Suppose that $X$ is a continuous random variable, and that $Y$ is a linear function ...
-1
votes
1answer
46 views

Exponential distribution and expectation [on hold]

Given that $X ∼ Exp(λ)$, compute $\mathbb{E}[e^{−(X−\lambda/2)^2} ]$. Your answer should not be left as an integral. so you would get $\mathbb{E}[e^{-x^2+x\lambda-\lambda^2/4} ]$? Can this question ...
0
votes
3answers
48 views

Probability on selecting balls

If I have B black balls and W white balls in a bag, what is the probability that the last one I select is white? How shall I solve this problem? I am not sure how to make a start, is it correct ...
1
vote
1answer
26 views

How to approximate the expected value in this problem

I was solving this probability problem and I don't know how to approximate the expected value. Thanks in advance! Problem definition: The durability of a tire in a city of South Africa is a ...
1
vote
1answer
43 views

Probability and Name of Distribution

Suppose $X$ has PDF $f_X$ given by \begin{align*} f_X (x) = \begin{cases} \frac{\alpha x_0^\alpha} {x^{\alpha+1}} &\text{if $x ≥ x_0$,}\\ 0 &\text{if $x < x_0$,}\end{cases} \end{align*} ...
1
vote
0answers
21 views

Transition density of an AR(1) process?

If we have an AR(1) process, i.e: $X_{t+1} = \alpha X_t + e_{t+1}$ with $X_0=0$ then what is its Markov Chain transition density? We know that for a Markov chain, the following holds: $P(X_{t+1}\leq ...
0
votes
0answers
31 views

2 people being born and dying on same dates with same names [on hold]

With 100 first names and 500 last names to choose from, odds of 2 people with same first and last names having exact dates of birth and death including year out of 100 million people? (this is a ...
0
votes
0answers
86 views

Write a random variable as a convex combination of other 2

I'm trying to prove that if $f:[0,1]\to\mathbb{R} $ is continuous and convex, then the Bernstein polynomials are too. The hint that I've got is this: "Let $p_1 < p_2 < p_3<1$ and consider ...
1
vote
1answer
23 views

Determine $P(S_n\leq1)$ where $S_n=\sum_{k=1}^nX_k$

Suppose that $X_n$ are i.i.d. $Uniform(0,1)$ random variables. Let $S_n=\sum_{k=1}^nX_k$ with $S_0:=0$. Then, determine $P(S_n\leq1)$. I know that maybe by using Characteristic function of $S_n$ ...
0
votes
1answer
38 views

Problem of Conditional Probability

I am learning Probability from Sheldon Ross book. One of the problems starts by giving the probability $P_N$ that there are no matches when $N$ people select from among their own $N$ hats as ...
1
vote
1answer
19 views

Expected value using indicator variables

Randomly, $k$ distinguishable balls are placed into $n$ distinguishable boxes, with all possibilities equally likely. Find the expected number of empty boxes. PROPOSED SOLUTION: Let $I_j$ be the ...
2
votes
1answer
30 views

Sufficient condition for $E(wu\mid v)=0$ given that $E(u\mid v)=0$?

I'm trying to figure out what condition concerning $w$ and $v$ would be enough for me to infer that $E(wu\mid v)=0$ given that I already know $E(u\mid v)=0$. Clearly, $w$ is a constant works: ...
-2
votes
2answers
62 views

I'm not able to solve conditional probability questions!! [on hold]

You are given: $\Pr(A) = {2\over 5}$, $\Pr(A ∪ B) = {3\over 5}$, $\Pr(B\mid A) = {1\over 4}$, $\Pr(C\mid B) = {1\over 3}$, and $\Pr(C\mid A ∩ B) = {1\over 2}$. Find $\Pr(A\mid B ∩ C)$ Okay, ...
-1
votes
1answer
34 views

Conditional Gambler ruin problem

A gambler repeatedly plays a game where in each round, he wins a dollar with probability 1/3 and loses a dollar with probability 2/3. His strategy is “quit when he is ahead by 2 dollars”, though some ...
0
votes
1answer
14 views

Inverse function for a sort of negative binomial distribution

I am trying to find the inverse function of $f(p) = \sum_{k=0}^{6}{\binom{6-H+k}{k} p^{7-H} (1-p)^k}$, where $0 \leq H \leq 6$ is a constant integer. Any ideas on how to do this? Or perhaps equally ...
0
votes
0answers
23 views

Quadrant probability of non-centric bivariate normal distribution

Suppose $(X,Y)$ has a bivariate normal distribuion with non-zero mean vector $\mu$ and covariance matrix $\Sigma$. What should $\mathbb{P}(X>0,Y>0)$ be? My attempt gives me an definite ...
1
vote
3answers
62 views

Gender Birth problem - Conditional probability [on hold]

A family has two children. Assume that birth month is independent of gender, with boys and girls equally likely and all months equally likely, and assume that the elder child’s characteristics ...