# Tagged Questions

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### probability question

I did a course on probability in undergrad, and I felt like brushing up on it by working out some problems a colleague gave me: "Two siblings are playing a dice game, where the winner is determined ...
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### Expectations of martingales

Consider a martingale $(M_n)_{n \geq 0}$ adapted to a filtration $(\mathcal F)_{n \geq 0}$ on a probability space $(\Omega, \mathcal F, P)$. Prove that, for each $k \leq n$; $$E(M_n M_k) = E(M_k^2)$$ ...
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### Two players $A,B$ throw two dice…

Two players $A,B$ throw two dice. A throw first, and they throw it in turns (i.e. $A,B,A,B,A...$). If $A$ gets sum of $10$ at the dice he wins, if $B$ gets $9$ - he wins. What is the probability ...
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### Proving a chain is aperiodic, and finding a stationary distribution.

We have an irreducible Markov chain with a not necessarily finite state space. It has a transition matrix $P$ such that $P^2=P$. Prove (1) the chain is aperiodic, and (2) prove $p_{ij}=p_{jj}$ ...
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### Question involving an invariant measure on a Markov chain

Suppose $\mu$ is an invariant measure for a Markov chain with state space $S$ with $\mu(i)p_{ij}=\mu(j)p_{ji}$ $\forall i,j \in S$. Describe a Markov chain with this property. Also, show that $\mu$ is ...
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### Bayes Theorem Drug Testing

A large company gives a new employee a drug test. The False-Positive rate is 3% and the False-Negative rate is 2%. In addition, 2% of the population use the drug. The employee tests positive ...
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### Tossing Dice repeatedly, probability that 2nd trial had more tosses than 1st one

John repeatedly tosses a die until a six occurs for the first time. Alice then repeats the experiment. What is the probability that Alice made more tosses than John?
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### I have a 10 jugs…( A problem at combinatorics)

In the first jug we have 2 red balls and 2 white balls. In each of the other jugs we have at 4 red balls and 4 white balls. We take a random ball from the first one and put it at the second jug, then ...
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### Is this a $\sigma$-algebra(closed under contable union)?

Could I say that this $$M=\{X\subseteq\Omega=[0,1):x\in X\iff y\in X\}$$ is an $\sigma$-algebra? I don't see whether it is closed under countable union. x,y are two singetons of $\Omega$ For ...
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### Clarify my understanding for central limit theorem from a statement

Asked what the central limit theorem says, a student replies, "as you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal". Is the student ...
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### Not countable generated sigma field

I need to show that $F=(A \in \Omega$| A countable or co-countable) with $\Omega$=(0,1] is not countable generated. I have started supposing that F is countable generated and I have a hont that tell ...
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### Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
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### We are giving $m$ prizes to $n$ people at lottery…

We are giving $m$ prizes to $n$ people at lottery... Question A: What is the probability that no one will get more then one prize (assume that $n\ge m$). Question B: What is the probability the ...
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### what the central limit theorem says

Asked what the central limit theorem says, a student replies, "as you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal". Is the student ...
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### Help with using the “Inclusion–exclusion principle”

I have question at probability that I need to use the "Inclusion–exclusion principle"... Hand of bridge is 13 cards that picked up randomly. What is the probability that we will have a King and Ace ...
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### FINDING PMF probability theory

A die is tossed until first 6 occurred. Let X be Random variable that number of one in the experiment. Find the PMF of X. And E(X). I noticed PMF of X is geometric distribution but I don't know why.
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### Help Finding a distribution for $2(5\bar{Y}^2 + Y_6^2)/U$

Let $Y_1, Y_2, . . . , Y_5$ be a random sample of size 5 from a normal population with mean 0 and variance 1 and let Y = (1/5) $\sum_{i=1}^5 Y_i$ . Let $Y_6$ be another independent observation from ...
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### Prove long-tailed distribution is heavy-tailed

Consider a sequence of iids $X_is$. I know a distribution is heavy-tailed if the $E(e^{tX_i}) = \infty$ for all $t>0.$. Additionally a distribution is long-tailed if ...
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### Show that if X has a density f such that f’ exists and is integrable?

Show that if $X$ has a density $f$ such that $f'$ exists and is integrable, then its characteristic function has the property : $\phi(t)=ο(t^{-1} )$ as $t\to \infty$. Hint: If $X$ has a density ...
### Sample Variance of Sample Mean is an Unbiased Estimate of Population Variance of Sample Mean,$\mathbb E[\mathbb v(\bar y)]=\mathbb V(\bar y)$.
I have a hypothetical data $2,3,4,5$. I have to draw sample of size $2$ and prove that : sample variance of sample mean is an unbiased estimate of population variance of sample mean, that is ...