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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1answer
24 views

Normality of Champernowne Constant

let $x= 0.1234567891011121314\ldots$ Let $N(k,n)$ be the number of occurence of the digit $k$ ($0 \leq k \leq 9$) in the first $n$ decimals of $x$. If $\lim_{n \to \infty} \frac {N(k, n)}{n}$ ...
3
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1answer
27 views

Determine how likely it is that a set of boolean data is produced by a distribution

Suppose we have a collection of independent Boolean random variables $X_i$ and $Y_i$ (for $1 \le i \le N$), and are told $p_i = P(X_i = 1)$ for all $i$. We are now given a set of values $x_i$ that was ...
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0answers
25 views

If $P(A) = 0.2$, $P(A \cap B) = 0.1$, $P[(A \cup B)'] = 0.3$, what is $P[(A \cap B) \mid (A \cup B)']$?

Suppose events $A$ and $B$ are such that $P(A \cap B)= 0.1$ and $P[(A \cup B)'] = 0.3$. If $P(A)=0.2$, what is $P[(A \cap B) \mid (A \cup B)']$? I tried solving it by using the conditional ...
2
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3answers
34 views

If $N_{\mid\Lambda = \lambda} \sim$ Poisson ($\lambda$) and $\Lambda \sim$ unif$(0,5)$, find the probability of zero occurring.

The number of storms in the upcoming rainy season is assumed to be Poisson distributed, but with a parameter $\Lambda$ that is also random and uniformly distributed on $(0,5)$. That is, $\Lambda \sim$ ...
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2answers
40 views

Suppose you roll two dice. Find the probability that you roll an 8, provided you roll a 7 or 8 first?

In the gambling game "craps," a pair of dice is rolled and the outcome of the experiment is the sum of the points on the up sides of the six-sided dice. The bettor wins on the first roll if the sum is ...
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3answers
33 views

Mrs. Grundy's Conditional Probability Problem

Mrs. Grundy has two children. Given that Mrs. Grundy has at least one child born on a Monday, what is the probability that both her children were born on Mondays? Assume that each child was equally ...
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2answers
43 views

What is the probability of consecutive heads on a biased coin in the favor of heads 80%? [on hold]

What is the probability of consecutive heads on a biased coin? The probability of heads on a single flip is 80%. Can you provide a formula for this? Also, after the first event has occurred, i.e. ...
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0answers
16 views

Nando de Freitas' Machine Learning Homework 2 Questions 1 & 2 Solutions

I've been following Nando de Freitas' Machine Learning course from UBC. While I have been enjoying the course I thought it would be good to see if I could do the homework along with it. So I'm on ...
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2answers
49 views

Picking randomly a number with two digits that can not be divided by $10$

Picking randomly a number with two digits that can not be divided by $10$ $A)$ Find the expectation value and the variance of the number $\color{gray}{\text{This part is for tommorow:}}$ My ...
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2answers
54 views

Different solution of probability problem from textbook

It is the problem 1.2.3 of Karlin's book An introduction to stochastic modeling: A population having $N$ distinct elements is sampled with replacement. Because of repetitions a sample of size r ...
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3answers
23 views

Marginalising out $B$ in $P(A \mid B,C)$

Let's say that I have $P(A \mid B,C)$ - is it accurate to say that $P(A \mid C)$ can be found like this: $P(A \mid C) = \sum_B P(A \mid B,C)$ I know the values of all $P(A \mid B,C)$ as well as ...
2
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0answers
40 views

What is the benefit of stochastic models over deterministic models? [duplicate]

I have posted a similar question earlier and I guess this sounds naive to all of you, but nonetheless let me just ask: Consider I have a simple and deterministic model $M$, with a number of input ...
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0answers
28 views

Help to understand the solution of the task about probability

The task is: The solution is: I can't understand how they got the first equation...
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1answer
36 views

Probability problem: Correct me if I am wrong.

There are two urns $U_1$ and $U_2$. $U_1$ contains four white and four black balls, and $U_2$ is empty. Four balls are drawn at random from $U_1$ and transferred to $U_2$. Then a ball is drawn at ...
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0answers
39 views

Proof of inequality 5 [on hold]

Let $B \subset F$ be a sub $\sigma$-algebra, inequality $|x||y|/\alpha\beta \leq x^2/2\alpha^2+y^2/2\beta^2$ with $\alpha\beta>0$ to prove $E\{|xy| \mid B\} \leq \alpha\beta$.
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0answers
32 views

Can this problem be posed in terms of Probability?

A photographer takes a picture of his vehicle's instrument panel, and later realized that the odometer reading == his zip code. Upon making this observation, he asks (figure of speech, really), "What ...
2
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1answer
32 views

Picking randomly one card till we get the first heart

From a standard pack of cards $(\spadesuit, \heartsuit, \diamondsuit, \clubsuit)$ we are picking randomly one card in a each time (with replacement) until we get the first heart $\heartsuit$. $(A.)$ ...
0
votes
1answer
54 views

Explain why $E(X)=1.65$ and $Var(X)=1.64$

Let $U$ be uniformly distributed on the interval [$\frac{1}{3},1$]. Let $X$ be a random variable such that the conditional distribution of $X$ given $U=p$ is Geometric with parameter $p$. (a) Find ...
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0answers
27 views

Probability of choosing the same color has the smallest probability on balanced set?

A bag contains $N$ balls of $K$ different colors. Suppose that there is, at least, $s>0$ number of each color. We would like to do the following procedure: Choose $s$ balls randomly (with ...
3
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2answers
39 views

If you roll a die two times, what is the probability the sum of the upturned faces equals $7$?

If you roll a die two times, what is the probability the sum of the upturned faces equals $7$? I can answer this question if I consider the order of the rolled numbers relevant. However, when I ...
2
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1answer
25 views

$27$ balls into $3$ cells

Spreading $15$ white balls and $12$ black balls into $3$ cells, each of which can contain any number of balls. $(A.)$ Find the probability that in each cell there will be exactly $5$ white balls. ...
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2answers
36 views

Root mean square distance explanation…

We know that $D_{rms}=\sqrt N$ where $N$ is the number of steps taken by the random walker. Now,consider a situation where a random walker walks $2$ steps in positive direction in the first two ...
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0answers
41 views

In a jug $3$ balls are white and $2$ balls are black

In a jug $3$ balls are white and numberd $0,1,2$ and $2$ balls are black numberd $0,1$ picking randomly $2$ balls (both togethet), let $W$ be the number of white balls that we picked , and let $N$ be ...
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0answers
16 views

Convergence in distribution (sufficient conditions)

Define a set of random variables $Z^2_i=\frac{(v_j-np_j)^2}{np_j} $ $K(Z_i,Z_j)=-\sqrt {p_i p_j}$ $E( Z^2_i ) = 1-p_i$ and $T_i= g_i - \sum_{j=1}^r g_j \sqrt p_j \sqrt p_l$ $K(T_i,T_j)=-\sqrt ...
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0answers
21 views

Neat expression for finite series with poisson distribution

I have the following expression $$ \sum_{n=1}^N f(k, n, p)\frac{1}{n} $$ where $f()$ is the binomial probability mass function: $$ f(k, n, p) = {n \choose k} p^k (1-p)^{n-k}$$ I wonder whether ...
0
votes
1answer
43 views

Probability of Binary Word Matching

Thanks for reading. Would be lovely if somebody could help me out on this (but not just post the answer) but also how you got there. I'm a programmer and I've ran across this problem which I can't ...
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0answers
15 views

Kelly criterion for Each-Way betting

3 outcome answered question Hi all, I've been having trouble finding the Kelly Criterion bet size for an each-way bet. The above link shows the solution to a problem with 3 distinct and mutually ...
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0answers
19 views

How are lottery winnings calculated?

I'm pretty familiar how most chance games payouts are calculated - the ratio shoul be inversely proportional to the probability of winning, minus house edge. If we bet the same amount on the same ...
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2answers
36 views

A priori probability in Bayesian inference problem

The problem A psychic uses a five-card deck to demonstrate ESP, claiming to be able to guess a card correctly with $0.5$ probability (of course, ordinary guessing is $0.2$). A single experiment ...
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2answers
49 views

There are three events: $A$ and $B$ and $C$. We know $P(A|B) = P(B|C) = 0.5$. Then $P(A|C)$ should be?

There are three events: $A$ and $B$ and $C$. We know $P(A|B) = P(B|C) = 0.5$. Then $P(A|C)$ should be? Is it $0.5\cdot0.5=0.25$? The question only provide the above information and the question ...
3
votes
1answer
37 views

Finding the Distribution of Y given $X_1 + X_2$ where X, Y ~ Poisson $\Lambda$

So, because this is honestly homework for a course, I'm primarily looking for a hint from where I've gotten so far. The question is very quick. $X, Y$ are independently distributed Poisson ...
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0answers
25 views

Is there a name for a “trivial” random variable defined on a finite probability space?

Let $(\Omega,2^\Omega,\Bbb P)$ be a finite probability space where $\Omega$ is a finite subset of $\Bbb N$. A "trivial" random variable $X$ st. $X(\omega)=\omega$ for any $\omega \in \Omega$ is ...
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0answers
22 views

Derive a CDF for a random variable based on its relationship with a second random variable? [duplicate]

Take a lifetime with the CDF $F(t)=1-(1-t)^n$ for $t$ in $[0,1]$ and some natural $n$. Now find the CDF of the variable $T_x=T-x$ when $T>x$ for $x$ in $(0,1)$. I need help getting started on this ...
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2answers
41 views

Find $\text{Var}(N)$ where $P(N = n|Y = y)$ is $\text{Possion}(y)$; $Y$ is a gamma with parameters $(r,\lambda)$

The question is as follows: Suppose that the conditional distribution of $N$, given that $Y = y$, is Poisson with mean $y$. Further suppose that $Y$ is a gamma random variable with parameters ...
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0answers
31 views

Finding MLE for $\mu^{2}$

The problem says the following: Let $X = (X_{1}, ..., X_{n})$ be a random sample, where $X_{i} \sim N(\mu_{0},1)$, where $\mu_{0} \in \mathbb{R}$ is unknown. I do not have problems calculating the ...
2
votes
1answer
42 views

Obtaining quadratic equation using Least Squares Method

This question is most likely extremely trivial, but I'm having some difficulty obtaining the least squares equation from the following data points: {{1.08, 0}, {1.07, 0.0659232}, {0.97, 0.1695168}, ...
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2answers
42 views

Finding the distribution of $n$ times the minimum of $n$ exponential random variables.

I'm having trouble with this question: Let $X = (X_{1}, \ldots, X_{n})$ be a random sample, where each $X_{i}$ is an exponential random variable with mean $\lambda_{0} \in (0,\infty)$, which is ...
2
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2answers
33 views

Find the CDF and density for the ratio $Z=X/Y$ given that $X$ and $Y$ are iid? [duplicate]

$X$ and $Y$ are iid r.v.s where $f(x)=\lambda e^{-\lambda x}$, $x>0$. We are given $Z=X/Y$ and asked to find the CDF and density for $Z$. I tried doing this using the multivariate transformation ...
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0answers
22 views

How can you picture Conditional Probability in 3D?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I modified the following's source for concision. $1.$ Now look at ...
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1answer
20 views

How can you picture Conditional Probability in a 2D Venn Diagram?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I pursue only intuition; do not answer with formal proofs. Which ...
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2answers
34 views

Picking random variables from [0,1] to calculate probability

I am trying to pick two points from $[0, 1]$ at random and with uniform probability. Let the result be the pair $(X, Y)$ in the square $[0, 1] \times [0, 1]$. Suppose that the distribution of $(X, Y)$ ...
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2answers
16 views

Math Test review on probability and statistics

A committee of 11 members is voting on a proposal. Each member casts a YES or NO vote. On a random voting basis, what is the probability that the proposal wins by a vote of 8 to 3?
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0answers
14 views

how to find value on piecewise cumulative density function

Consider f(x) = k/x^2 on x≥1 and 0 elsewhere. I am trying to find k. I took the integral from 1 to infinity of k/x^2 and got a result of 1. I know this is wrong because the next step, I try to find ...
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0answers
22 views

Find the conditional and marginal densities?

I'm getting lost in notation here. We have $f_X(x)=xe^{-x}$ and we have a new random variable, $Y$, which is uniform from $(0,X)$. We want three things: Marginal density for $Y$ Conditional ...
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1answer
21 views

symmetrical random walk P[M(n)=k]

On a symmetrical random walk, I am trying to deduce P[$M_{n}$ = k] = $(\frac{1}2)^n$ ${n \choose \frac{n+k}2}$ where n is the total number of steps and ${n \choose \frac{n+k}2}$ is the number ...
0
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1answer
16 views

Different sign in solution on probability problem

It is the problem 1.2.2 of Karlin's book Introduction to stochastic modeling: Let $N$ cards carry distinct numbers $x_1, x_2, ..., x_n$. If two cards are drawn at random without replacement show ...
1
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1answer
29 views

Compute the probability that of the $10$ years with the highest temperatures over the last $100$ years, $9$ occurred in the last $11$ years

“From data on global average temperatures over the last $100$ years, it turns out that of the $10$ years with the highest temperatures, $9$ occurred in the last $11$ years.” Compute the probability ...
5
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1answer
219 views

Hillary Clinton's Iowa Caucus Coin Toss Wins and Bayesian Inference

In yesterday's Iowa Caucus, Hillary Clinton beat Bernie Sanders in six out of six tied counties by a coin-toss*. I believe we would have heard the uproar about it by now if this was somehow rigged in ...
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1answer
16 views

Find the marginal and conditional densities without explicitly having the joint density?

Take a random variable $Y$. You're given $f_Y(x)=xe^{-x}$ when $x>0$. Given $Y$, a new random variable, $M$, is uniform over the interval $(0,Y)$. I need to calculate the marginal density for $Y$, ...
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0answers
18 views

Prove $V(x)$ is an increasing function (involving PDF and CDF)

I need to prove the following: $V(x) = x + G(x)/g(x)$ is an increasing function where $G(x)$ is a CDF and $g(x)$ is the corresponding pdf. When I take the derivative, I get $$1 + g(x)^2/g(x)^2 - ...