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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3
votes
0answers
37 views

Knight (Chess) Problem on telephone keyboard

There is phone keyboard with Knight on 0 (as shown below). 123 456 789 0 Knight moves as per the rules of chess (2 straight and one turn). T is no. of moves ...
1
vote
2answers
33 views

How to compute variance of a conditional expectation and vice versa

I am trying to use the law of total variance which is $$\operatorname{Var}(X) = \text{Var}(E(X\mid Y)) + E(\operatorname{Var}(X\mid Y))$$ But I honestly have no idea how to compute either one of ...
0
votes
0answers
14 views

Variance of a discrete random variable with 4 outcomes

Say we have an event with four possible outcomes $(1,2,3,4)$ each with equal probabilities of occurring. A random variable $Y$ is defined as the number of outcome 1's that occur over $n$ events. ...
2
votes
1answer
49 views

Which of the following is true for random variables $X$ and $Y$?

Suppose $X$ and $Y$ are two random variables such that $aX+bY$ is a normal random variable for all $a,b \in \mathbb{R}$. Consider the following statements P, Q, R and S: (P) : $X$ is a ...
0
votes
0answers
29 views

Probability and Decision Tree

The task in hand is the following: There are N entities (E1..EN). There are M questions (Q1..QM). Each question has at least two options (can be different number of options of different questions). ...
0
votes
0answers
25 views

Concentrated Likelihood

I am working on Panel Data models and I am having some issues to obtain the concentrated log-likelihood function for the following model. $$ y_{it} = \delta'x_{it} + \beta_i'f_t + u_{it} , i = 1,...,N,...
0
votes
0answers
16 views

Expected Value of Geometric Distribution, with Exponential Random Variable [on hold]

Given $X~Geometric(1/10), E(X) = (1-1/10)/(1/10) = 9$. What is $E(X^4)$?
0
votes
5answers
45 views

What is the probability that both cards are not aces?

Suppose two cards are drawn from a standard 52 card deck without replacement. Assuming all cards are equally likely to be selected, what is the probability that both cards are not aces? My Solution ...
1
vote
0answers
15 views

Asymptotic Growth of Markov Chain

I'm interested in the following problem: We have got a time-discrete Markov chain $(X_n)$ with state space $S=\mathbb{R}_+^d$. The transition kernel is discrete in the sense, that for each $s \in S$ ...
3
votes
1answer
63 views

The Uncountable and Probability

Suppose we draw a random uniformly number from $[0,1]$, if we do this countable many times, how many times will we get $1$, I suspect $0$? If we do it uncountable many times, how often will we get $1$?...
-1
votes
0answers
18 views

Log-Poisson distribution. Literature?

If I have a Poisson distributed random variable $X$, then adding two to get $Y=X+2$ I can define the new rv $Z=\log(Y)$. I would call this a log-Poisson distribution or better exp-Poisson ...
1
vote
1answer
46 views

How to calculate the probability $P(2\leq X<4)$ from the distribution $F(X)$.

Let the random variable $X$ have the distribution function $$F(x)=\begin{cases}0 , &\text{if }\ x<0\\ \frac{x}{2}, & \text{if }\ 0\leq x<1\\ \frac{3}{5},&\text{if }\ 1\leq x<2\\ ...
0
votes
1answer
18 views

Probability question confusion

A bag consists 20 cards contains 4 cards of each Colours, red , green , blue , yellow and white . The four cards of Each Colour are numbered 1,2,3,4 respectively . One card is selected at random . ...
0
votes
2answers
21 views

Probability of two statistically independent, uniformly distributed variables occurring within time frame of each other?

Say two events will occur independently of each other, only once each. The time of each event occurring is uniformly distributed from 0 to 10 seconds. What is the probability that the events will ...
0
votes
0answers
35 views

Conditional Expectation

If we have $Z_{t-1} = (X_1, \ldots,X_{t-1})$ and we know $(S_{t-1} \mid Z_{t-1}) \backsim N (\hat{S}_{t-1}, P_{t-1})$, where $S_t = G_{t}S_{t-1} + w_{t}$ and $X_{t} = F_tS_t +v_t$, so I know that $(...
1
vote
2answers
18 views

Number of outcomes, if having known the distinct numbers and number of choices

This question came to me, when I was solving another relavent question in my class: We have $N$ distinct numbers, say $P(X=i)=1/N$, with $i=1,...,N$. We choose $n$ (known) numbers from them (with ...
1
vote
0answers
96 views

Joint probability with constraint [on hold]

Let's say that one is conducting an experiment with 8 units and 4 units have to be assigned to treatment. Assuming all units' respective treatment assignment probabilities are greater than 0 and less ...
0
votes
0answers
25 views

n-dimensional Archimedean copulas

I am studying Nelsen's book Introduction to copulas and I want to extend $\max(1-[(1-u)^{\theta}+(1-v)^{\theta}]^{1/\theta},0)$ to an $n$-dimensional copula. The problem is that it seems that the $\...
1
vote
1answer
21 views

Linear estimation of an exponential distribution

QUESTION We have $Y \sim \mathrm{Exp}(1/6)$. We define $T = e^{−4Y}$ Calculate the best linear estimator of $T$ according to $Y$ ANSWER Ok it sounds pretty simple at first I got my $f_Y(y)=\frac{...
0
votes
2answers
42 views

How to correctly count the probability for a computer game situation? [on hold]

Imagine we have the following situation in a computer game: One player has two minions with 30 and 6 hitpoints correspondingly. Another player casts a spell which does 12 times 1 damage (for each of ...
-4
votes
0answers
38 views

What is E(X^a)? [on hold]

In terms of expected value, is there a formula for $E(X^a)$, such that a is any real number? If not, how does one do so knowing the distribution of X, using the formula $E(X) = \sum xp_x(x)$?
1
vote
0answers
61 views

For Which General Distributions Does This Inequality Hold?

Let $X$ be a random variable with mean $\mu$, where $0 < \mu < 1$. Let $X(n)$ be the sum of $n$ independent ,identically distributed, $X$ variables. Under what conditions on $X$ , possibly ...
0
votes
0answers
25 views

Distribution for playing n scratcher lottery tickets

If I know the prize distribution for a scratcher lottery ticket (i.e. the various prize amounts and the probability associated with each prize) is there a way to form a distribution for playing, say, ...
0
votes
4answers
81 views

$E(X)$ versus $E(X|Y)$

Why is $E(X)$ considered a constant but $E(X|Y)$ considered a random variable? Seems like confusing notation since I'd assume the latter is a fixed constant "the expected value of random variable $X$ ...
0
votes
3answers
53 views

Mapping a PDF to a uniform distribution on $(0,1)$

Let me preface this by saying that I'm not familiar with differential equations, other than basic "separable" differential equations. This problem has come up in a Probability problem that I am doing. ...
0
votes
1answer
32 views

Find $E(Y)$ and $Var(Y)$ of $\log Y \sim N (\mu,\sigma^2)$

Find $E(Y)$ and $Var(Y)$ of $\log Y \sim N (\mu,\sigma^2)$ I tried solving this in 2 different ways. The second way is what I am stuck on: 1st Way: Let $Y=e^X$ where $X \sim N (\mu,\sigma^2)$. ...
0
votes
1answer
34 views

How to check if an estimator is expectation right?

I am sorry if the terminology is a little bit off, and anyone that knows the correct terminology please correct me. Let us assume we have 3 independent measures, X1, X2, and X3 from the same ...
1
vote
2answers
33 views

Is there a book or lecture notes on Percolation Theory containing exercises?

I have seen Grimmett's Percolation Theory and I have also seen a few online lecture notes. But they don't have exercises. I understand it is stupid to ask of exercises in such a recent and hot ...
1
vote
0answers
22 views

Probable number of missing records

[Note: This is a duplicate of a post I made in Stack Overflow] I am having difficulty grasping a probability assumption in a problem I am reviewing. Given: each record in a dataset has a unique ...
0
votes
1answer
29 views

Proof that normalized vector of Gaussian variables is uniformly distributed on the sphere

I have seen in various places the following claim: Let $X_1$, $X_2$, $\cdots$, $X_n \sim \mathcal{N}(0, 1)$ and be independent. Then, the vector $$ X = (\frac{X_1}{Z}, \frac{X_2}{Z}, \cdots, \frac{...
1
vote
1answer
45 views

Probability generating function of some “random walk”

Let $S_n=\sum^n_{r=0}X_r$ be a left-continuous random walk on the integers with a retaining barrier at zero. More specifically, we assume that the $X_r$ are identically distributed integer-valued ...
0
votes
1answer
26 views

Expected value of a point/dot to be near the closest side of a square surface?

If we choose a random point inside a square of side length 1, what is the probability for the point is nearest a chosen side of the square? I would say that the answer is 1/4 since there are 4 sides ...
-2
votes
1answer
17 views

Bayesian probability calculation [on hold]

You have one fair coin, and one biased coin which lands Heads with probability $3/4$. You pick one of the coins at random and flip it three times. It lands Heads all three times. Given this ...
0
votes
1answer
41 views

Specifying from the general in probability: Does it work? [on hold]

If the average classroom AC holds 30 students, and 1 in 10 students throughout the US has a probability of having condition A, does that mean there's a 300% chance there's a student in classroom AC ...
0
votes
1answer
15 views

Exponential distribution solving X^2, lambda value known

Let $X$ be a random variable with exponential distribution with parameter $\lambda=2$. The expectation of the random variable $Y=X^2$ is equal to a) 1/2 b) $\sqrt{2}/2$ c) 1 d) 2 e) 4 I've ...
2
votes
1answer
108 views
+100

Expected score from threshold

We play a game where a sequence of $n$ numbers is drawn uniformly from $[0,1]$, and we need to set a threshold $0\leq a\leq 1$. For every number that is at least our threshold, we get $a$ points, ...
0
votes
1answer
168 views

Probability - Discrete Distribution

A roulette wheel has $38$ numbers. Eighteen of the numbers are black, eighteen are red, and two are green. When the wheel is spun, the ball is equally likely to land on any of the $38$ numbers. Each ...
2
votes
1answer
39 views

Showing that if $X \sim \operatorname{Exp}(1)$, then $Y = F_X(X)$ has uniform distribution on $[0,1]$

Let $X \sim \operatorname{Exp}(1)$, and show $Y = F_X(X)$ has uniform distribution on $[0,1]$. I calculated $F_Y$, since the cumulative distribution function identifies a distribution. We have: $$\...
2
votes
1answer
55 views

Expectation of Product of Ito Integrals wrt Independent Brownian Motions

Let $W_1(t)$, $W_2(t)$, $W_3(t)$ be independent Brownian motions and $f$, $g$ smooth functions. I want to know if the following is true: $$ \mathbb{E}\left[ \left( \int\limits_0^t f(...
1
vote
3answers
40 views

Expected Value for Heads for Unknown Weighted Coin Given Head First Flip

This is a combinatorics problem, and I think it involves expected values and conditional probability, but I don't know how to use them: "A bag contains an infinite number of coins whose probabilities ...
0
votes
1answer
39 views

distributing numbered balls with duplicates into 4 boxes [on hold]

How many ways are there to distribute 52 balls, numbered 1 to 13 with 4 duplicates for each number, into 4 distinguishable boxes.
0
votes
1answer
15 views

The expected number of rallies needed to break a deuce in table tennis / tennis

In table tennis & tennis (among other sports?), a player/team must be two points ahead of their opposition to win a game. Thus, a game could technically go on for infinite deuces. However, ...
1
vote
2answers
14 views

Dependency of function of independent random variables

$X$ and $Y$ are independent and identically distributed random variables, $c$ is a constant. I wonder if $\frac{1}{X+c}$, and $\frac{1}{Y+c}$ are independent? In other words, are the functions of ...
1
vote
1answer
38 views

Poisson Distribution - Optimistaion

A store offers a new seasonal product featured. Let $N$ be the random variable which means the number of clients who come to the store during the season, where $N \sim \operatorname{Poisson}(19)$. ...
1
vote
3answers
66 views

Law of total expectation?

Apparently $E[X] = E[E[X\mid Y]]$ but I don't understand what this really means. I looked at https://en.wikipedia.org/wiki/Law_of_total_expectation but need another explanation. Isn't this the same ...
2
votes
2answers
76 views

Is there an intuitive meaning of $p - p^2$ [closed]

If $p$ is the probability of an event occurring, does $p - p^2$ have an intuitive meaning?
1
vote
4answers
45 views

Bayesian probability problem?

Problem: In a city there are three types of taxis which drive towards the airport. 30% are blue, 20% green, 50% yellow. They take there customers too late with probabilities 0.1,0.2,0.3 respectively. ...
0
votes
0answers
4 views

How to compute the probability and CI of replicating multiple previously observed statistically significant p-value?

The FDA often requires a sponsor to conduct multiple clinical trials prior to approval. Given the following observations in a ph2 and ph3 trial, how would you go about predicting the probability of ...
4
votes
1answer
57 views

Proving $\text{Var}(X) = E[X^2] - (E[X])^2$

I want to understand something about the derivation of $\text{Var}(X) = E[X^2] - (E[X])^2$ Variance is defined as the expected squared difference between a random variable and the mean (expected ...
43
votes
12answers
10k views

If a coin toss is observed to come up as heads many times, does that affect the probability of the next toss?

A two-sided coin has just been minted with two different sides (heads and tails). It has never been flipped before. Basic understanding of probability suggests that the probability of flipping heads ...