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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8 views

On $[0,1]$ 100 points are chosen at random. $X_1$- number of chosen point between $(0, \frac{1}{5})$

On $[0,1]$ $,100$ points are chosen at random. $X_1$- number of chosen points between $(0, \frac{1}{5})$ $X_2$- number of chosen points between $( \frac{1}{2},1)$ Find: CDF of $X_1, X_2, ...
3
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1answer
17 views

A die is thrown 10 times. What is the probability that $6$ isn't registered and that at least one “1” is registered.

$A$, first occurrence - that $"6"$ isn't registered $B$, second occurrence - that at least one $"1"$ is registered. What I know: How to find $P(A)$ and $P(B)$ (over their complements) What I'm ...
4
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1answer
23 views

This is a variation of the colored socks in a drawer problem.

Suppose that instead of having one drawer, you have two drawers. Each drawer has some socks that are white and some that are black. Drawer 1 has w black socks and x white socks. Drawer 2 has y ...
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1answer
20 views

How to find double probability with monte carlo simulation in R

$$ f(x) = \begin{cases} C\exp(-\frac{1}{2}x^3), & \quad x >-1,\\ 0, & \text{othewise}. \\ \end{cases} $$ Here, $c=2.2702$ and I want to find the probability of ...
1
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1answer
26 views

Let $q_n, n=1,2,3,\ldots$ be on stringing of rational numbers from $[0,1]$.

$a.) $Let $q_n, n=1,2,3,\ldots$ be on stringing of rational numbers from $[0,1]$ and let the be given a sequence of sets: $A_n=[q_n,1] n=1,2,3,\ldots$ Find $$\overline{\lim_{n\to \infty}} A_n.$$ Now ...
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1answer
22 views

Why do characteristic functions use $e^{ix}$ and not $e^{-ix}$? Does it matter?

I've heard the characteristic function be described as the Fourier-Stieltjes Transform of the distribution measure of a r.v., but I was curious as to why it's written as $E[e^{ix}]$ and not the ...
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2answers
22 views

A problem about probability.

The question is: "Maria flip a coin for $6$ times while Davide for $7$ times. What is the probability that Davide obtains more heads than Maria?" I solved this problem analysing $7$ cases: $1)$ ...
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2answers
33 views

Manufacturing problem, exponential distribution

A manufacturing process produces $92%$ good chips (G) and $8%$ bad chips (B). The lifetime, in seconds, of chips is exponentially distributed $E(\lambda)$.For good chips, ...
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0answers
22 views

Calculate the expectation of psi(x) with a gamma density x∼Gamma(α,β )

Suppose I have a Gamma distributed random Variable x∼Gamma(α,β). Now I want to calculate the following expectation values (integrals): E[psi(x)] with psi(x) being the digamma function Many thanks ...
-3
votes
1answer
29 views

Variance of a special random walk [on hold]

I am trying to find the variance of the following special random walk: Suppose that $U=(U_1,U_2,...)$ is a sequence of independent random variables, each taking values $u$ (for up) and $d$ (for down) ...
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2answers
24 views

Conditional probability question 4

I'm studying for my finals and I'm stuck at this question. I know the solution for a) is 10/13 and b) is 2/3, I just wasn't able to get there. Any help would be greatly appreciated. Here it is: In a ...
3
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1answer
25 views

an exercise about changing the measure and convergence in $L^1$

this is exercise 17.12 from probability essentials written by jacod & protter. Suppose $lim_{n→∞} X_n = X$ a.s. Let $Y = sup_n |X_n − X|$. Show $Y < ∞$ a.s. , and define a new probability ...
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1answer
19 views

Showing $X$ has finite expectation if $Y$ has finite expectation and $P(|X-Y| \leq M ) = 1$

Suppose $X$ and $Y$ are two random variables such that $$P(|X-Y| \leq M) = 1 $$ for some constant $M$. Show that if $Y$ has finite expectation, then $X$ has finite expectation and $|EX - EY| \leq M,$ ...
1
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1answer
13 views

Number of moves necessary to solve a generalized fifteen-puzzle with random moves

Consider the famous fifteen-puzzle, but with size $m\times n$ ; $m,n\in \mathbb N$; $m,n>1$ Suppose, the initial position of the puzzle is random but solveable. Random moves are made until the ...
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0answers
24 views

Deal with circular standard deviation

I have a sensor that gives me a direction of the target in degrees with a certain standard deviation and I use a circular probability density function (PDF) to represent probable locations for the ...
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1answer
8 views

Relationship between univariate normal distribution and multivariate normal distribution

Let $a_1, a_2, a_3$ is column vector and $H = [a_1 a_2 a_3]$. If $a_i$ have standard normal distribution, is this following statement true ? $$ vec(H) = [(a_1)^T (a_2)^T (a_3)^T]^T$$ have multivariate ...
2
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4answers
37 views

Is the probability of a chain of dependent events, independent of the order in which they occur?

So I have a bag of 6 balls: 3 red, 2 blue and 1 green. Events: A: I draw a red ball B: I draw a blue ball C: I draw a green ball I do not replace the balls (thus, resulting in conditional ...
0
votes
4answers
45 views

Rolling two dice, what is the probability of getting 6 on either of them, but not both? [on hold]

Rolling two dice, what is the probability of getting 6 on one of them, but not both?
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0answers
20 views

Probability of boys participating in at least two of the sports

At a school for boys, there are $240$ students in Grade 12. A survey was done to ascertain the boys' involvement in sport. $120$ play Rugby (R). $60$ play Soccer(S) . $95$ play cricket (C). $17$ Play ...
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1answer
14 views

What is the probability of selecting the correct key from a basket? [on hold]

Nine women and two men sat in 11 chairs from left to right. A male occupied seat 9 and 11. Keys were drawn from a basket in the order they sat. There were 11 keys in the basket and one key is drawn at ...
0
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1answer
28 views

Let $X: U(0,1)$ and when $X=x$ then $Y:U(\frac{x}{2}, \frac{2x}{3})$ uniform distribution. Find the density function of $Y$ and $EY$

Let $X: U(0,1)$ and when $X=x$ then $Y:U(\frac{x}{2}, \frac{2x}{3})$ uniform distribution. Find the density function of $Y$ and $EY$ I don't know if it would be presumptuousness to say that $Y: ...
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0answers
13 views

How do I find $ Pr\{X_1 < k \} $ and $ Pr\{X_1 > k \} $if $X_1 : G(p_1)$- geometric distribution

I would think the song like $1-Pr\{X_1 < k \} $ but what is confusing to me is the fact that this is a discrete random variable, and these inequalities ussually apply to absolute continuous ...
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vote
2answers
21 views

Drawing marbles out of a bag with or without replacement

Suppose the marble population has $n$ marbles. And based on prior knowledge I know that 30% of them are red, 50% of them are green, and 20% of them are blue. If I want to sample 2 marbles from my ...
-1
votes
1answer
16 views

Trinomial Distribution - Cumulative Probability of (X-Y) [on hold]

In an election there are n voters. They can each vote for Candidate A (with probability p); Candidate B (with probability q) or neither (with probability (1-p-q) ). What is the Probability that ...
2
votes
0answers
38 views

Probability calculation involving order statistics

There are $n+1$ independent and identically distributed random variables with the same distribution as $D \sim \text{Exp}(\mu)$, denoted by $D, D_1, D_2, \ldots, D_n$. Define event $E_1$ as "$D$ is ...
0
votes
2answers
17 views

Find constants using the mean, variance and covariance of two random variables

Given two random variables $X$ and $Y$ such that $\mathbb{E}(Y\mid X)=a+bX$, find $a$ and $b$ in terms of the mean, the variance and the covariance of $X$ and $Y$. Hint: What is the relationship ...
2
votes
1answer
24 views

Find the probability generating function $G(s)$ of this branching process.

Suppose that $X_n$ is size of the $n$th generation of a branching process started from a single individual, where each individual has a random number of children with probability mass function: ...
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votes
1answer
24 views

The Crawfords have $3$ children &mdash; questions about their probability distribution [on hold]

Suppose the Crawfords have three children. Assume the probability of a boy or a girl is $\frac12$ for each birth. How many possible outcomes are there? What is the probability that only two of the ...
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votes
1answer
31 views

Time taken to give answer if probability is given.

This is a question that I am struggling with: Since the password is periodically changed, you would like to know the answer as soon as possible. So you decide to interrogate the minions in an order ...
1
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2answers
18 views

Probability of Independent Events individual vs in series

I understand that independent events (such as a fair coin flip) should not be viewed in succession. For example, if you flip heads 10 times in a row, the odds of flipping the next coin heads is still ...
1
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1answer
28 views

Show that the empty set is independent of $A$ for any $A$

I am somewhat stumped as to how to approach this. The only thing I can remotely think of is $$P(A\cap \emptyset) = P(\emptyset)$$ but nothing else comes to mind. Suggestions?
2
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0answers
23 views

Convergence of Sum of Random Variables “Independent in Limit”

Consider a sequence of random variables $X_n\sim U[-n,n]$ and a random variable $Y\sim N(0,1)$, all independently distributed. In addition, consider a bounded, measurable function $f:\mathbb{R}\to ...
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1answer
20 views

Suppose that $N$ is an iid geometric RV and $X_i$ is an iid Bernoulli RV. Find the p.g.f. of $R=X_1+ \dots + X_n$.

Each year a tree of a particular type flowers once and produces a random number $N$ of flowers, where $\mathbb{P}(N=n)=(1-p)p^n$, $n=0,1,2,\dots $ and $0<p<1$. Each flower has probability $1/2$ ...
4
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0answers
44 views

How many possible shuffles can be won perfectly?

It is known that the possible shuffles of a deck of cards is $52!$, or ~$80658175170943878571660636856403766975289505440883277824000000000000$ different combinations. I have become aware of a game ...
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0answers
13 views

Expectation Value and Generalized Holder Inequality

In the context of probability, I need help in interpreting a generalized Holder inequality (wiki): $\| \prod_{k=1}^n f_k \|_r \leq \prod_{k=1}^n \| f_k \|_{p_k}$, where $\sum_{k=1}^n \dfrac{1}{p_k} ...
0
votes
1answer
30 views

Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following :

$$P(X_1 < X_2 < X_3)= \frac{(1-p_1)(1-p_2)p_2p_3^2}{(1-p_2p_3)(1-p_1p_2p_3)}$$ To be frank I do not know where to start with this question, I would like an idea to get me going, or better yet ...
2
votes
1answer
20 views

Convergence in probability of product random variables

If $Y_n$s converge to constant $c$ in probability & $(X_n)$ is a sequense of random variables, is it true that $X_nY_n- cX_n$ converge to $0$ in probability? How can I prove this? Thanks in ...
2
votes
2answers
26 views

Conditional probability calculation (multivariate distribution)

X and Y are two i.i.d. random variables having the uniform distribution in $[0,1]$ The question is to calculate $Pr(Y\geq \frac{1}{2} | Y\geq 1-2X)$ My calculations: $$ \begin{align} Pr(Y\geq ...
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1answer
20 views

How to derive the covariance formula

In my book, I am given this proof: $$ Cov(X,Y) = \mathbb{E}[X - \mathbb{E}X][Y - \mathbb{E}Y]$$ $$ Cov(X,Y) = \mathbb{E}[XY] - 2\mathbb{E}[X] \mathbb{E}[Y] + \mathbb{E}[X] \mathbb{E}[Y]$$ I do not ...
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1answer
22 views

multinomial hypothesis testing

Suppose we have data $(X_1, X_2, X_3)$ (I'll refer the categories as 1, 2, 3) that has a multinomial distribution with parameters $n$ and $(p_1, p_2, p_3)$ and we want to test the hypothesis that ...
2
votes
2answers
39 views

Probability and Combinatorics without replacement

If I have a sample space of $A$ and I randomly select $a$ elements, mark them, put them back into the sample space, then randomly select $b$ elements and I want to know what the probability is that ...
0
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1answer
22 views

Expected time of drawing all types of coins from a large pile

I've been working on the following question but am uncertain of how to solve it Consider an infinitely large pile of coins. Each coin has a number {1, 2, . . . , n} written on it, and these numbers ...
2
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0answers
30 views

An exercise on conditional expectation and some related questions.

I tried to solve an exercise involving conditional expectations, and in doing so some question's popped up in my mind. First the exercise: $|Z| \le c \textrm{ P.-a.s.} \Rightarrow |E\{ Z | ...
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2answers
30 views

A stick length 1 is broken into 2 pieces. Let $Z_1$ be the length of the shorter part. Find $EZ_1$

This is used: If $p(x)$ is continuous, then $P\{x \leq X \leq x+ \Delta x \}= p(x)\Delta x+ o(\Delta x), \Delta x\to 0.$ Let $H_1$ be the occurrence that the point at which the stick is broken is in ...
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1answer
27 views

Find the probability that this laptop is produced by the firm B. [on hold]

The store received three laptops of different firms (firm A, firm B and firm C) in a ratio of 6:1:3. Laptops coming from firm A do not require repair during the warranty period in $98$ percent of ...
0
votes
1answer
26 views

Finding $E[W]$ and $E[W^2]$, where $W = \int_{t=0}^T B_s$ $ds$

I'm trying to find a)$E[W]$ and b) $E[W^2]$, where $W_t = \int_{t=0}^T B_s$ $ds$ ($B_s$ denotes a Brownian motion). In addition, I'd like to find $E[Z_sZ_t]$, where $Z_t = \int_0^t B_s^2$$ ds$ ...
1
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1answer
19 views

Joint probability distribution (over unit circle)

A couple of two continuous random variables $(X,Y)$ is distributed uniformly over the closed unity circle (so $-1\leq x \leq 1$ , $y$ analog). $U$ is defined as the distance from $O$ to the point ...
2
votes
1answer
35 views

Translation:Bayes Classificator -> precise math?

I want to understand the most simple form of the Bayes classificator (see here) but I want to understand it in a really precise, clean, mathematical way. Math description of the setting: Let us ...
0
votes
1answer
22 views

Take the outcome of a draw in ELO formula

Is there any way to get the probability of a draw outcome using ELO formula as it only gives the Win probability ELO formula is given by $E = \frac{1}{1+10^\frac{d}{a}}$ where d is the difference in ...
0
votes
1answer
34 views

Expectation of the product of Brownian motions

I'm new to Stack Exchange. I'd like to find the expectation of the product of three Brownian motions: $E(B(t_1)B(t_2)B(t_3))$ I know from a previous post here that ...