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 ...

learn more… | top users | synonyms (1)

0
votes
0answers
12 views

Random Variable Problem with unrestricted Parameters Worded Problem

I have no idea how to go about solving (a) -> (c) For (a) Is K=0.2 because k/1-0.8=1 Hence, P(Z=z) = 0.2(0.8)^x But How do we determine the mean or variance with unrestricted z values.
0
votes
2answers
17 views

Random Variable Worded Problem

I can figure out the basics to the question, that is the mean and variance of Y: E(Y) = 1-2p Var(Y) = 4p(1-p) I don't understand parts (i) and (ii). I dont understand the question itself, that ...
0
votes
1answer
11 views

Expectation of Random Variable - Probability Worded Problem

The part I am confused with is (c) I found part (a) which is: p(0) = 7/24, p(1) = 21/24, p(2) = 7/40 and p(3) = 1/120 How do we find the values for a and b, for part (c) ?
0
votes
1answer
26 views

Expanding the expected value

How to expand: $E(Y+1)^2$ my working out: $E(Y^2)+E(1^2) = E(Y^2)+1$ (I'm not sure why this is though..) Can someone link to or list the rules for expanding the expected value ......
0
votes
2answers
34 views

If probability density functions converge a.e., then cumulative density functions converge

I have read a conclusion in a textbook: Suppose $f_n,f$ are density functions of some r.v. also $f_n\to f$ a.e., then $$\int f_n \mathrm{d}x \to\int f \mathrm{d}x $$ Fisrt I want to use ...
-1
votes
1answer
28 views

Sinchronicity and probability [on hold]

"Synchronicity is the experience of two or more events as meaningfully related, where they are unlikely to be causally related. The subject sees it as a meaningful coincidence." Wikipedia First, I am ...
1
vote
2answers
16 views

Finding values of a constant in a probability distribution

A probability distribution for the random variable $X$ is defined by: $$\mathbb{P}[X=x] = K\cdot(0.9)^x,\quad x = 0,1,2,\ldots$$ It is asked to find $\mathbb{P}[X\geq 2]$. When there is a domain for ...
0
votes
0answers
11 views

Density function of $(X_{(1)}, X_{(2)})$ where $X_i$ ~ Exp(1)

I'm reading an answer to a question, which quotes a (non-English) book that I don't have access to. I'm wondering if someone here can shed some light on this claim: Let $X_1, X_2, ...$ independent ...
2
votes
2answers
39 views

Maximizing heads/number of flips game

Flip a coin until you wish to stop. Your goal is to maximize the ratio number of heads/total number of flips. What is the expected value of this game? Additionally, how would one play this game?
0
votes
0answers
13 views

Multivariable chain rule and an integral function

Can the chain rule be used in the following scenario? Let $u(r,t) = \int g(r,t)~f_R(r)~dr$ where $f_R(r)$ is a well-defined probability distribution. Then we can write $u(r,t)=u(x,y)=\int x~y~dr$ ...
2
votes
1answer
28 views

Find some probabilities given the probability tree

i've been practicing probability since it's not my strength, but i am doing that without a tutor or an official course, just books and videos. I was reading a problem, and i was capable of draw the ...
0
votes
0answers
9 views

Is it possible to use multiple time scale algorithm here?

Suppose a random sequence is being generated (the next term generated depends on the previous term, but we don't know any distribution) until we hit some specific number. We want to calculate the ...
1
vote
1answer
33 views

Expectation of uniform distribution with unknown parameter, given maximal (minimal) observation.

Let $x_i \text{ be} ~ i.i.d. ~ \sim Uni[0,\theta]$ $(\theta \text{ unknown})$. Denote $M_n = \max x_i$. So, through circumferential means, I can show that $E(x_1|M_n) = \frac{n+1}{2n} M_n$. The ...
0
votes
0answers
13 views

Ross probability models questions [on hold]

I am studying for a course and have no professors to talk to live, so I hope some members here can be kind enough to help me. Rather than writing everything out, and splitting it up into different ...
2
votes
1answer
24 views

Proof that a random variable has exponential distribution.

Supose that $X_1$ is a continuous and positive (real) random variable with exponential distribution, namely $$P(X_1>t)=e^{-\lambda t}\quad t>0$$ Now suppose that $X_2$ is another continuous and ...
1
vote
0answers
26 views

Probability for sparse matrix after permutation

I need to calculate the probability of the following question, it is kind of tricky but I cannot work out the exact value. For a sparse (most of its entries are zero) matrix $X=[x_1, x_2, \cdots, ...
1
vote
1answer
35 views

If the two-engine plane cannot take off unless both engines are operating properly, which plane is safer on takeoff?

I am practicing a bunch of probability problems I find through random sources and I am stuck with this one. Suppose the probability that the engine in a single-engine fighter will fail on take-off is ...
1
vote
3answers
57 views

Plausibility vs Probability

http://whatho.in/2013/plausibility-versus-probability/ refers to pp 155-156 of 533 of Thinking, Fast and Slow by Daniel Kahneman. I'll use one of Kahneman's other questions from p 156: A ...
3
votes
2answers
68 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
2
votes
2answers
65 views

Brainteaser: Player A has £1, Player B £99. They flip a coin. The loser pays the other £1. Expected number of games before one is bankrupt?

Player A has £1, Player B £99. They flip a coin. The loser pays the other £1. What is the expected number of games they play before one is bankrupt? I have struggled at this for hours now with little ...
9
votes
2answers
349 views

What is the expected value of the number of circles formed?

There are $99$ identical square tiles, each with a quarter-circle drawn on it. When the tiles are randomly arranged in a $9$ by $11$ rectangle, what is the expected value of the number of full circles ...
0
votes
1answer
40 views

How is this paper using probability notation?

I am trying to understand this paper about documents and sentences. At the end of page three, they say: Let g(wi, wj ) be the distance between two events (1 if in the same sentence, 2 in neighboring, ...
6
votes
2answers
42 views

What is the expected value of the number of anchors of $S$?

For any subset $S\subseteq\{1,2,\ldots,15\}$, call a number $n$ an anchor for $S$ if $n$ and $n+ |S|$ are both elements of $S$. For example, $4$ is an anchor of the set $S=\{4,7,14\}$, since $4\in S$ ...
5
votes
2answers
54 views

What is the expected value of A?

The Happy Animals Kennel has 18 cages in a row. They allocate these cages at random to 6 dogs, 6 cats, and 6 pot-bellied pigs (with one animal per cage). All arrangements are equally likely. Let A ...
0
votes
0answers
21 views

Expectation of a logarithmic/trigonometric function

I am trying to find a closed form solution of the following expectation: $$\mathbb{E}[\log(a+b\cos(\phi))]$$ where $a$ and $b$ are real constants, and the expectation is with respect to $\phi$. If ...
1
vote
1answer
24 views

Fudge Dice: Reroll vs. Bonus

A "fudge die" is a die with equal probability to result in -1, 0, or +1. The commercially produced fudge dice are generally 6-sided dice with two "–", two "+", and two blank sides. In the ...
0
votes
1answer
22 views

CDF of random variables

due to my lack of knowledge in probability theory, I have first to apologize if the following question is not formulated in a proper language. I was wondering if there is any formal expression of the ...
-2
votes
1answer
33 views

Mean of max vs max of mean

If I have say an $n$ collection of 10 random variables $X_1, \ldots, X_{10}$ (so an $n \times 10$ matrix of values) from some underlying distribution whether Gaussian or uniform, and I calculate ...
7
votes
10answers
2k views

Should I throw the dice again if I have rolled 4?

My math skills are very basic so it might be a stupid question, I had a discussion with my brother in law and now we have a 'math problem'. We were playing a game with dices and he threw 4. The ...
1
vote
1answer
38 views

Find one-dimensional distribution function $F(y\mid t)$ of random process $Y(t)$

$ Y(t)=tZ^2;\quad Z\sim U(-2;2); \quad t\ge0. \quad$ I need to 1) find one-dimensional distribution function $F(y|t)$ of random process $Y(t)$. 2) calculate probability that trajectory of the ...
2
votes
2answers
94 views

10 little dwarves

A dwarf-killing giant lines up 10 dwarfs from shortest to tallest. Each dwarf can see all the shortest dwarfs in front of him, but cannot see the dwarfs behind himself. The giant randomly puts a ...
0
votes
0answers
29 views

Probability of infinite intersections

While I was studying Probability and random processes I came across the following question. Say I have $A_1,A_2, \ldots, A_n$ events such that $A_i$ is in $E$ but not equal to $E$. What is: ...
1
vote
1answer
33 views

A Bayesian estimate of the bias of a coin

Consider a coin with unknown probability $p$ of landing on head. I will toss the coin and stop as soon as I get a head. Say this is after $n$ tosses. If my prior belief for $p$ was uniform on ...
1
vote
3answers
26 views

Probability of getting a certain group of students when choosing three at random out of 25

A teacher randomly chooses a group of three students from her class of 25 students. Find: a) Probability that friends Suri, Lily and Violeta are chosen for the group? b) If he ...
3
votes
2answers
48 views

Parity of the sum of consecutive Bernoulli random variables

$\newcommand{\Var}{\operatorname{Var}}$I have $X_1,X_2,\ldots,X_{n+1}$ i.i.d. rv, each $X_i$ is a Bernoulli rv with parameter $p$, i.e. $X_i \in \{0,1\}$, $P(X_i=0)=1-p$ and $P(X_i=1)=p$ with $0 \leq ...
1
vote
2answers
83 views

How to obtain probability distribution from the generating function $G(s) = e^{a(s-1)^2}$?

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...
0
votes
1answer
34 views

Integrability condition

Suppose that \begin{align} \mathbb{E}\int_{0}^{T}f^{2}(t)dt <K \end{align} Does it also hold that \begin{align} \int_{0}^{T}f^{2}(t)dt <K \end{align} ? Here, T, K>0 are fixed. I am a bit ...
0
votes
1answer
51 views

Maximum likelihood estimators

I have $X_1,X_2,\dots,X_n$ as random samples from a binomial distribution, with probability function: $$p_X(x) = Pr(X=x) = {m \choose{n}}\alpha^x(1-\alpha)^{m-x},x=0,1,2,\dots,m$$ where $m$ is given ...
1
vote
4answers
74 views

Estimate bias of a coin

Consider a coin with probability $p$ of landing on head. You can estimate the prob by tossing it lots of times and looking at the proportion of heads one gets. In my problem I just want to know if ...
-2
votes
1answer
15 views

Standard deviation of travel times

Suppose that travel times for Swinburne students are normally distributed with mean of $32.5$ minutes and a standard deviation of $5$ minutes. Complete the following sentence, giving figures correct ...
2
votes
2answers
38 views

Probability problem: n different balls in n different boxes

Problem Suppose $n$ different balls are distributed in $n$ different boxes. Calculate the probability that each box is not empty when distributed the balls. I'll define the sample space as ...
2
votes
1answer
27 views

Invariance Properties of Brownian Motion

I am trying to make sense of the Scaling-Invariance and Time-Inversion properties of Brownian motion by producing a sample path. For the record, I am using the following definitions. Let $B(t)$ be the ...
1
vote
3answers
81 views

Probability that the red fish are the first species to become extinct

I have a doubt in the solution of the next problem: A pond contains $3$ distinct species of fish, which we will call the Red, Blue, and Green fish. There are r Red, b Blue, and g Green fish. ...
1
vote
2answers
20 views

probability that the white balls are left in the urn

I don´t understand the solution of next problem: An urn contains n white balls and m black balls. The balls are withdrawn one at a time until only those of the same color are left. Show that with ...
1
vote
1answer
41 views

Probability of drawing the king of hearts and a red card

Two cards are drawn from a standard deck of cards at the same time. Find: a) Probability of drawing the King of hearts and a red card b) Probability of drawing the King of hearts and a black ...
0
votes
0answers
12 views

Check work for finding Max log-Likelihood of a geometric Distribution

Here is my geometric distribution: $P(L=n)=p^{n}*(1-p)$ To find the max likelihood, I do: $\sum_{L_i} L_i\log(p) + \log(1-p)$, where L_i is a particular length. I take the derivatives and end up ...
0
votes
1answer
71 views

Why is the expected average displacement of a random walk of N steps not $\sqrt N$?

Let $D_N$ be the expected average of the displacement of a random walk on $\mathbb Z$ from the origin, where $N$ is the number of steps, each of which is either $-1$ or $1$. We take the definition of ...
-2
votes
0answers
10 views

Does $\sum_B p(L|B)p(B|G) = \sum_B p(L,B |G) = p(L|G)?$ [on hold]

Does $\sum_B p(L|B)p(B|G) = \sum_B p(L,B |G) = p(L|G)?$ By chain rule, does $p(L,B |G) = p(L|B,G)p(B|G)$? Does $\sum_B p(L|B)p(B|G) = \sum_B p(L|B,G)p(B|G)$
1
vote
2answers
69 views

what is the difference between average and expected value?

I have been going through the definition of expected value in Wikipedia (http://en.wikipedia.org/wiki/Expected_value) beneath all that jargon it seems that the expected value of a distribution is the ...
1
vote
1answer
22 views

Conditional probability with a normal distribution

Given that Y and L are normally distributed, the expectation of L given Y is $\mu (Y)$ and the variance of L given Y is $\sigma ^2 (Y)$, why is the conditional probability $P(L > x| Y) = \Phi ...