Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

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30 views

Show $P(X > x) = e^{-\lambda x} \forall x > 0$ and some $\lambda > 0$

Given that $X$ is a non-negative random variable, satisfies $P(X = x) = 0$ and $P(X > x + y\mid X > x) = P(X > y)$ $ \forall x, y \in \mathbb{R^{+}}$. Prove that $P(X > x) = e^{-\lambda ...
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16 views

Joint distribution integral limits for marginal

Given a joint distribution as: $$f(x,y)=\frac{21}{4} \cdot x^2 y \,\,\text{ where } x^2\le y\le 1 $$ I have problem finding the limits for $x$ to do the integral to find marginal of $y.$
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1answer
27 views

What is the statistical probability to the following real world problem?

It's been a while since I last took statistics so I need help verifying my calculations to a real-life practical problem. My wife and I want to do IVF with PGD to get a girl. The OBGYN at the ...
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0answers
20 views

we use the poisson random distribution

Let ܺ denote the number of typographical errors on a single page of the lecture notes. Let’s assume that ܺ has Poisson distribution with parameter ߣ = 1. Calculate the probability that there is at ...
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1answer
19 views

how can we derive both the CDF of $M$ and the PDF of $M$

Given $X_i \sim Uniform[0, 1]$ for $i = 1, \dots, n$. What is the distribution of $M := \min(X_1, \dots, X_n)$? I feel like I'm missing something and I've been stuck on this for two days
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19 views

Show that 𝑌1, 𝑌2, … are the arrival times of an inhomogeneous Poisson process. What is the intensity of this Poisson process?

Successive offers for my house are independent, identically distributed random variables $X_1, X_2$, ... having density function $f$ and distribution function $F$. Let $Y_1 = X_1$, let $Y_2$ be the ...
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1answer
17 views

A general expression for the sum of multiple independent Normal Mixture Distributions?

Suppose random variable $X_1$ is a mixture of two Normal distributions with means of $\mu_A$ and $\mu_B$ respectively, standard deviations of $\sigma_A$ and $\sigma_B$ respectively, and weights given ...
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1answer
36 views

How many ants are expected to pass through the hole before it closes?

A long line of ants find a hole in the pantry door. At time zero the first ant enters the pantry and then, one after the other, the ants pass through the hole at a rate of exactly one every minute. ...
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1answer
25 views

Finding the limit probability of sample mean greater than a number and its distribution

Given that $X_{1}, X_{2}, X_{3}...$ is squence of independent random variable, given that $E[X_{n}] = \frac{20}{17}$ and $E[|X_{n}|^3] < 12$, let $\overline{X_{n}} = \frac{X_{1} + X_{2} + ... + X_{...
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1answer
15 views

Does the following sequence of R.V. converge (L^1 and a.s.)

Consider the sequence of random variables $(X_n)_{n \in \mathbb{N}}$ on the probability space $((0,1],\mathcal{B}((0,1]))$ defined by $$\begin{align*} X_1(\omega) &:= 1_{\big(\frac{1}{2},1 \big]}(\...
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25 views

Non-standard CLT problem

$X_1, X_2, X_3,...$ independent r.v.'s and $\mathbb{P}(X_n=\frac{-1}{\sqrt{n}})=\mathbb{P}(X_n=0)=\mathbb{P}(X_n=\frac{1}{\sqrt{n}})=\frac13$ check if $$\frac{X_1+X_2+X_3+\dots +X_n}{\sqrt{\log(n)}}$$ ...
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10 views

minimising mean square error of an estimator [duplicate]

$X_1...X_n$ is a random sample from a normal distribution N($\mu,\sigma^2$). now to estimate $\sigma^2$ we use an estimator of the form: $C_n\sum_{i=1}^n(X_i-\bar{X})^2$ and $C_n$ depends on sample ...
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1answer
16 views

Show dependence of unity of unit squares

Let $(X_1, X_2)$ be a pair of uniform distributed random coordinates on $S:=([-1,0]\times[-1,0])\;\cup\; [0,1]\times[0,1])$. Are $X_1, X_2$ independent? Are $X_1, X_2$ uncorrelated? If you draw the ...
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1answer
10 views

Determining probability that test vector different from reference vector

I have a reference vector, with a mean and standard deviation for each of its n elements. I want to compute the probability that a test vector is different than the reference vector. For a single ...
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1answer
17 views

Is the following rv integrable?

Consider the probability space $((0,1],\mathcal{B}((0,1]),\lambda|_{(0,1]})$ and define $$X_n(\omega) := \frac{1}{\omega} 1_{\big(0, \frac{1}{n}\big]}(\omega).$$ I am told that this R.V. is not ...
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1answer
26 views

Number of succes S is a Binomal variable $n=50$ and $p=0.75$

Approximate(with normal curve and correction for continuity) the probability that S is bigger then 39? Approximate Pr(S=40) and compare it with the exact calculation? R can be used
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29 views

Overall chance to win in Monty Hall problem variation, when you don't know about the game is it really 2/3? [closed]

50% odds it is game number 1 where host avoid to open your door in first round even if there is no award 2/3*1/2=2/6 = 4/12 1/3*1/2=1/6 = 2/12 50% odds it is game number 2 where host dosen't avoid ...
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1answer
17 views

Do these conditions imply weak convergence of the random variable?

Here is a question from a past exam from probability theory that I try to tackle: Let $X,X_1,X_2,\ldots$ be real random variables. We know that: (a) $X_n^2$ converges in distribution to $X^2$ (b) $...
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1answer
24 views

Expected value of the longest run of heads or tails in N flips of a coin

What is the expected value of the longest run of heads or tails observed in $N$ flips of a fair coin? I'd like to consider this for any $N$, small or large.
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2answers
25 views

Number of original toys found after buying $k+1$ packs of milk.

I was trying to play around with the Coupon Collector's Problem and got to solve this related problem: There is a promotion in the store: you get a free random toy per 1 pack of milk. The collector ...
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0answers
12 views

Proof of Strong Law of Large numbers in Course of Prob Theory

I am trying to follow a proof of the SLLN in the textbook a course in probabilty theory. Theorem: If the $X_j$ are uncorrelated and their second moments have a common bound then the SLLN follows: ...
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2answers
13 views

What is the most probable number of acceptable screens in the next batch of 10 screens and what is the probability?

A novel process of manufacturing laptop screens is under test. In recent tests, it is found that 75% of the screens are acceptable. What is the most probable number of acceptable screens in the next ...
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2answers
25 views

Normally distributed rain drops problem

About 50% of raindrops land downtown, downtown is a perfectly circular space around the city centre. Assuming the coordinates of the raindrops are independent and distributed according to the standard ...
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2answers
13 views

Given the value of random variable $S_1,$ what is the best prediction $g(S_1)$ for the value of $S_2$?

Suppose $$ S_1 = \exp(X_1), \quad \quad X_1 \sim N(\mu_1, \sigma_1^2) $$ $$ S_2 = \exp(\lambda X_1 + X_2), \quad \quad X_2 \sim N(\mu_2, \sigma_2^2). $$ Assume $X_1$ and $X_2$ are independent and $\...
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1answer
12 views

Variance of a compound random variable.

Let $Y$ be such a random variable, that : $Y = \begin{cases} 1, & \mbox{if } X \le 1/2 \\ X, & \mbox{if } X>1/2 \end{cases}$ where $X$ has uniform distribution on $[0,1]$. My solution: ...
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1answer
26 views

bound of random variable implies bound of conditional expectation?

Given two dependent real random variables $X,Y$, suppose that \begin{equation} Y \leq c \Rightarrow X \leq k \end{equation} and that everything is well defined. I would like to argue that \begin{...
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24 views

derivative a convolution formula

Let X and Y be independent exponential random variables with mean 1. I'm trying to derivative a convolution-like formula for $Z = XY$ but I'm having troubles getting started. The convolution ...
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2answers
28 views

Likelihood function for MLE

Refer to the Example 7 in this lecture: How did the author obtain the likelihood function ? Is it from binomial? Can someone show the steps to the likelihood function? Thank you!
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1answer
18 views

Generalized Jensen's equality on averages

Let $y,\,z:[0,1]\rightarrow \mathbb{R}$ be integrable functions. $y\left(p\right)$ is weakly increasing and positive ($0 < y\left(p\right)\leq y\left(p'\right)$ for $0 \leq p\leq p'$). Assuming ...
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0answers
24 views

Tossing a fair coin and denote his probability

I am trying to calculate the probability of an infinitely tossed coin. The exercise is: A coin is tossed infinitely often, where the probability of success is $p$. The variable $Y_k$ is $1$, if the ...
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1answer
15 views

compute the conditional expectation given by following random variables

Consider the probability space $([0,1],\mathcal B[0,1],\Lambda)$ where $\Lambda$ is Lebesgue measure. Let $\xi$ and $\eta$ be two random variable . For any set $A$, denote by $\Bbb I_{A}(x)$ the ...
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68 views

Is this distribution already known and has a name?

I am asking whether the distribution on $\Bbb R$ with probability density $$ f(x) := \frac 2 {\sqrt{2\pi}} e^{-\frac{x^2}{2}} - 2 \vert x\vert \int_{\vert x \vert}^\infty \frac 1{\sqrt{2\pi}} e^{-\...
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1answer
12 views

Probability in dice throws with external property.

The question: Q rolls a fair 6-sided dice and tells you the result. Q lies 3/4th of the time. If Q chooses to lie he will always pick 6, unless he got a 6, in which case he will pick a different ...
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1answer
16 views

Prove weak convergence (proof verification)

We are given a distribution, $\mathbb{P}(Y_n=\frac k n)=2^{-k}$ for $k= 1, 2, 3,\dots$. Check if this converges in distribution and if it does find the limit distribution. I think it does, the CDF ...
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16 views

How to intuitively understand convergence in distribution?

I have this question: Let $\{X_n\}$ be positive integer valued random variables. Prove that $X_n\xrightarrow{d}X_0$ iff for every $k\geq0$, $P[X_n=k]\rightarrow P[X_0=k]$. The answer: Suppose $...
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1answer
13 views

Show the equality of E[Z] = $\sum_{k=1}^{n} P[M_K]$ (see the task)

i am trying to show the equality but i dont get any further with the exercise. Can someone give me a hint? Here is the task: Let $A_1,A_2...,A_n$ be events ($A_k \in \mathcal{F}$) a) Express the ...
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12 views

Size of 2 Clusters in a Uniform distribution

Let we have two uniformly distributed cluster points X and Y in a square of Area 1. There are other ...
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2answers
63 views

Stuck on a probability problem/Expectation of coin toss

I'm stuck on the following problem that is due for tomorrow: We're flipping 1 coin indefinitely. $X$ is a random variable that count the amount of coin tosses. What is the expectation of the ...
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0answers
17 views

Poisson Formula finding lamina?

A microprocessor manufacturing facility produces 300 microprocessors per hour. The probability that an individual chip is faulty is 0.01. Calculate the probability that in a given hour's ...
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3answers
45 views

Suppose that $X$ is a random variable, prove that $\operatorname{Var}(X) \geq 4.5$

Suppose that $X$ is a random variable for which $\mathbb E(X) = 10$, $\Pr(X \leq 7) = 0.2$, and $\Pr(X \geq 13) = 0.3$. Prove that $\operatorname{Var}(X) \geq 4.5$. I'm not really sure how to start ...
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0answers
46 views

Checking a bound on the Stieltjes transform from Terence Tao's notes

I'm trying to check a bound on Stieltjes transform of a probability measure, that's given in equation (2.92) on P. 170 in Terence Tao's notes "Topics in Random Matrix Theory". Denote the Stieltjes ...
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1answer
42 views

Convolution of two uniform

Let $X$ be a uniform random variable on $[0,1]$, let $Y$ be uniform in $[3,5]$ independent of $X$. Find the probability density function of $X + Y$. My solution is: $$ (f_X * f_Y)(x) = \begin{cases} ...
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2answers
18 views

Show $ \mathbb{E}(T_1\mid N_1=0)=1+\frac{1}{\lambda}$ for a Poisson process

For a Poisson process with rate $\lambda$, show $ \mathbb{E}(T_1\mid N_1=0) = 1 + \frac{1}{\lambda}$. attempt We know $T_1 \sim \operatorname{Exp}(\lambda)$. The solution I'm looking at says to ...
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1answer
41 views

Determine the mathematical expectation of the number of games in such a match [closed]

The probability of victory of the younger brother over the elder is 4/7 in each party (there are no draws), and the results of all parties do not depend on each other. They play a match until one of ...
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19 views

Question about probability with replacement

Two marbles are drawn at random from a box containing 3 red (R) and 2 blue (B). Find the probability that the two marbles are of different colors if two are drawn w/ replacement. What I've tried is ...
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2answers
40 views

Markov chain problem to write a recursive equation

write a recursive equation for $a_N(i)$ by considering what happens on the first transition out of state $i$. Please help me on this problem. I don't know how to start. Thanks!
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1answer
23 views

Probability of $9$-digit permutation with every $5$-digit subsequence divisible by $3$ or $5$

A nine-digit number is formed using the digit $1,2,3,4,5,6,7,8,9.$ Find the probability of forming a number such that product of any of its $5$ consecutive digits is divisible by $3$ or $5$ What I ...
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24 views

Calculate number of defects(PMF of sum of multinomial distributions)

Suppose you have two factories, A and B which produce two items at the same time with a probability of defect from A, $p_A$ and probability of defect from B, $p_B$ Suppose a total of $N_T$ items are ...
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0answers
11 views

Combination of groups of like objects into a set [closed]

There are $5$ red, $4$ yellow, $3$ green, $2$ blue, and $1$ black ball. $5$ balls are picked randomly without replacement $A)$ How many combinations of $5$ balls are possible $B)$ What is the ...
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2answers
34 views

Almost Sure Convergence of a Series of Random Variables

Let $\psi(x) = x^2$ when $|x| \leq 1$ and $\psi(x) = |x|$ when $|x| \geq 1$. Show that if $X_1, X_2, \dots$ are independent with $\mathbb{E} X_n = 0$ and $\sum_{n=1}^\infty \mathbb{E} \psi(X_n) < \...