Tagged Questions

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Finding the probability of given model

I'm trying to model a system given as a project, any help or advice is useful for now. Here our assumptions for a part of it: If I study for the finals and do assignments, I have $80\%$ chance to ...
43 views

A question regarding the central limit theorem

I would appreciate if you could please take a look at the following attempt. I am preparing for an upcoming exam, and this is taken from a past exam paper. I would like to know if it is correct. Thank ...
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satellite radios has a 3% defective rate. A random sample of 1000 radios is drawn. What is probability the defective rate is greater than 5%?

The assembly line that produces satellite radios has historically resulted in a 3% defective rate. A random sample of 1000 radios is drawn. What is the probability that the defective rate is ...
49 views

Mean and variance of call center data

I have a fairly involved homework question, I was wondering if I could get some help. There are two types of phone calls arriving at a switch, long-duration and short-duration. Each day the number ...
56 views

Moment generating functions question

Suppose that $X_1, X_2, ..., X_n$ are independent, where each $X_i$ has probability (mass) function $p_i(x_i)$ given as $p_i(x_i) = \frac{e^{-\lambda} \lambda_i^{x_i}}{x_i!}$ (only the parameter ...
198 views

Probability the sum of numbered balls drawn from a box is odd

A box contains 100 balls, numbered from 1 to 100. If 3 balls are selected at random and with replacement from the box, what is the probability that the sum of the 3 numbers on the balls selected from ...
88 views

Using the Bayes Theorem?

A certain town has two taxi companies, the Green Taxi Co (cars coloured green) and the Blue Taxi Co (cars coloured blue). 10% of taxis are the Green and 90% are the Blue. There was an accident ...
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calculate joint Probability density function limit distribution of statistics $T_n=(nU_{(1)},nU_{(2)}).$

Suppose $U_{(1)} , \dots , U_{(n)}$ is order statistics a random sample from $U(0,1)$.how can find joint Probability density function limit distribution of statistics $T_n=(nU_{(1)},nU_{(2)}).$
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Maximum Likelihood Estimator (MLE)

Salmon sometimes carry a parasite anisakis simplex which they pick up when feeding on krill at sea. The number of parasites on each fish might be assumed a random variable X having a probability ...
70 views

Normal approximation to binomial

What do I do when the normal approximation is not valid? Here's the question I'm trying to answer: A student guesses all 15 answers on a multiple-choice test. There are 5 choices for each of the ...
97 views

Why are the Borel subsets on $\mathbb R$ a $\sigma$-algebra?

I am a newbie in mathematical statistics and haven't learnt any group theory before. My lecture notes are too brief. How can the Borel subsets on $\mathbb R$ satisfy A.2 and A.3 of a $\sigma$-algebra ...
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Probability of 2 uniformly randomly generated, n-digit numbers (n>=4) having the same last 4 digits?

I apologize if question is too simple! For example, is the probability of generating 2, 8-digit numbers that have the last 4 digits match the same as the probability of the last 4 matching with a 12 ...
193 views

How does variance change as sample size increases

Situation: n possibilities each have their own probability of happening, and their own payout when they do. So expected payout $E_n$ is $\sum\limits_{i=1}^n \text{probability}_i*\text{payout}_i$ ...
122 views

Bayesian inference with Gaussian distributions

This is Problem 4(c), Chapter 2 from Thrun's Probabilistic Robotics . Note that this is self-study and not homework. Suppose I know my position $x$ to be a normal distribution with density ...
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May I get HMM calculations Checked?

I was trying to understand Hidden Markov Model(HMM). I was working out some examples. The first work out was on initial probability, transition probability and emission probability. I was trying to ...
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Are my HMM calculations going fine? [closed]

I was trying to understand the hidden Markov model (HMM) and to do some calculations, and I got some doubts. I attach my study in this Google Drive File. Can you check if my calculations are fine? I ...
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Probability proofs using ordered samples

Given an ordered i.i.d sample $X_{(1)}, \dots, X_{(n)}$ from a continuous distribution $F(x)$. How can it be shown that: (1) $\text{Pr}(X_{(k)} \leq x) = \text{P}r(N(x) \geq k)$ where $N(x)$ is the ...
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How many measurements (10 million molecules each time) are needed to analyze 15 million unique molecues in a 20 million pool?

This is my homework: There are 20 million DNA molecules in a library for high throughput DNA sequencing, each sequencing run can generate 10 million reads (i.e., analyze 10 millions of DNA molecules), ...
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Conditional probability questions (homework)

I copied the exact question. Can anyone check my answer? I am so confused between the choices, all of them seems to be workable. An company divides their customers according to how long they take ...
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Normal Distribution [duplicate]

Using the normal distribution. Let $X\sim \mathcal{N}(1, 2)$ and $Y\sim \mathcal{N}(2, 3)$ where $\mathcal{N}(μ, \sigma^2$) denotes the normal distribution with mean $\mu$ and variance $\sigma^2$. X ...
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Finding the z score and p-value of a binomial distribution

Emily is a big fan of lady gaga, and 20% of the songs on her ipod are lady gaga songs. Suppose Emily has her ipod on shuffle and repeat mode, which can be assumed to mean that each song to be played ...
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Why doesn't this represent a normal approximation to the binomial?

Suppose the registrar's office at a college reports 58% of the students live on campus. An intern working in the administration building is unaware of this 58% parameter value. He designs a study in ...
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probability of one random variable being greater than another

Using the normal distribution. Let $X \sim N(1, 2)$ and $Y \sim N(2, 3)$ where $N(\mu, \sigma^2)$ denotes the normal distribution with mean $\mu$ and variance $\sigma^2$. $X$ and $Y$ are independent. ...
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Probability of picking a biased coin

Suppose you have a bag of 100 coins of which 1 is biased with both sides as Heads. You pick a coin from the bag and toss it three times. The result of all three tosses is Heads. What is the ...
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Is the skew normal distribution log-concave?

Is the skew normal distribution log-concave?
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How to derive the conditional posterior density in hierarchical bayesian models?

I was reading on Gelman's Bayesian Data Analysis - Chapter 5 - Hierarchical model Suppose: data : $y_j$ s parameter: $\theta$ hyperparameter: $\phi$ On page 126, he mentions the analytical ...
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How to set up a posterior predictive test quantities (Bayesian context) to check for independent Poisson distributions?

Suppose we are given data $y_j \sim \text{Poi}(\lambda)$ and assume $y_j$ are iid. We can assume the prior distribution for $\theta$ follows $\text{Gamma}(\alpha, \beta)$, The posterior ...
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Realizations of random variable

Can some of you help me with following exercise? Provide a procedure to generate realizations of a random variable with CDF (cumulative distribution function) $Fx(x)$ given by: ...
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Average waiting time

who can help me to resolution of this statistic exercise? below the track: Caio go in a bank,the number of customers ahead him are described by a Poisson random variable of parameter a>0. Calculate ...
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How to compute the PDF of a sum of a discrete and a continuous random variable? [closed]

I have a problem with this exercise in probability and statistics: Calculate the probability density function (PDF) of $$Z=X+Y$$ where $Y$ is discrete random variable which is equal to $-1$ and $1$ ...
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Convergence in distribution and CDF

Suppose $X_n$ converges in distribution to $X$ , $x_n \rightarrow x$, also the cumulative distribution function for $X$ is continuous at $x$. Show that $P(X_n \leq x_n) \rightarrow P(X \leq x)$. ...
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Calculate a tennis players chance to win a 5set match knowing he has 50% chance of winning 1SET?

Is the following correct? My friend tells me i need to (add up the binomial probabilities of 2 wins in 2 sets, 2 wins in 3 sets and 2 wins in 4 sets THEN multiply the sum obtained by p) P(3) = 5C3 x ...
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Wigan scores after 30 minutes. Calculate home, away and draw in percentage terms with poisson regression

I have the following exercise: Wigan v city 90 mins match: Wigan scores after 30 minutes. Calculate home, away and draw in percentage terms with poisson regression, assuming pre-match odds ...
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Find the limiting distribution of $W_n$

Having a little trouble with this one: Suppose $X_1, X_2, \ldots$ are iid standard normal random variables. Let $W_n = \sqrt{n} \frac{X_1 + \cdots + X_n}{X_1^2 + \cdots + X_n^2}$. Find the limiting ...
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How to find the joint density of 3 normal variables?

Suppose $U, W, V, S$ are four independent normal random variables with mean $0$ and variance $1$. Let $X=W+U$, $Y=2W+S$, $Z=3W+V$. What is $f(X, Y, Z)$? Thanks!
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How to find principal components from variance-covariance matrix?

If I have the following var-cov matrix: $\Sigma_{A,B,C} = \left(\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \\ \end{array}\right)$ (1) How can I find a constant a, ...
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My question is about a homework question that I found interesting. It gives another proof (without using martingales) for that the critical Galton Watson tree dies out eventually. But it has given a ...
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Probability that at least one person at a party will accidentally choose their own gift

Thirty people are invited to a holiday party, where a gift exchange will take place. Each person brings a gift to put into the pot, and at the end of the night each person selects—at ...
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Given that $Y$ follows multivariate normal distribution ,i.e, $N_n (0, \sigma^2 I_n)$, we want to find the distribution of $Y'Y$ given that $a'Y=0$ where $a$ is a non zero constant vector. I know ...