Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis.

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Functions of sufficient statistics - are also they sufficient?

Are $1-1$ functions of sufficient statistics (for a given parameter) also sufficient for that parameter? For example, if $X$ is a sufficient statistic for parameter $p$, is $\log(X)$ also sufficient ...
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233 views

Kolmogorov-Smirnov two-sample test

I want to test if two samples are drawn from the same distribution. I generated two random arrays and used a python function to derive the KS statistic $D$ and the two-tailed p-value $P$: ...
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42 views

Mutual information expressed as Kullback-Leibler divergence

My lecturer defines the mutual information: $$ I(X;Y\mid Z) = D_{KL}\big(p(X,Y\mid Z)\parallel p(X\mid Z)\;p(Y\mid Z)\big)$$ Is this correct? I feel like it doesn't really make sense to say that; ...
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238 views

Chebyshev's inequality for 1 standard deviation results in 0?

In applying Chebyshev's inequality to a probability distribution, the following is the given equation: $$p(\mu - c*\sigma \le X \le \mu + c*\sigma) \ge 1 - \frac{1}{c^2}$$ This indicates for any ...
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Covariance between $X$ and $Y$ of a bivariate normal distribution?

$X$ and $Y$ have a bivariate normal distribution with $\sigma_X$= 5 mL, $\sigma_Y$= 2 mL, $\mu_X$= 120 mL, $\mu_Y$= 100 mL, and $\rho$ = 0.6. How do I find the covariance of $X$ and $Y$? I know the ...
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60 views

Method of moments estimation for $\theta$

I read one example in my notes, but I couldn't find out how the answer in my notes is derived. If $x_1,...,x_n$ are realizations of a random variable distributed with the following PDF: $f(z; \theta)...
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175 views

Finding distribution of distance from origin

A shot is fired at a circular target. The vertical and horizontal coordinates of the point of impact (taking the centre of the target as origin) are independent random variables, each distributed ...
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218 views

If X and Y are independent exponential random variables with respective parameters λ1 and λ2, how do I find the distribution of Z = X/Y?

If X and Y are independent exponential random variables with respective parameters λ1 and λ2, how do I find the distribution of Z = X/Y ?
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Find the Indicated Probability

Question: On one tropical island, hurricanes occur with a mean of 2.74 per year. Assuming that the number of hurricanes can be modeled by a Poisson distribution, find the probability that during the ...
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49 views

Question regarding the conditional probability of multiple events

I'm having trouble really understanding a seemingly easy question about conditional probability. Here is the question: Let $n_{R}$ denote the number of red balls in an urn and $N$ denote the number ...
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Find the indicated probability?

Question: A spinner has equal regions numbered 1 through 15. What is the probability that the spinner will stop on an even number or a multiple of 3? How do I begin to solve this?
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product of densities

One can frequently read, that the product of the densities of two INDEPENDENT random variables is also a density - the joint density of the two variables. (see for example: http://en.wikipedia.org/...
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Very simple question about subsets

Is {1, 2, 2, 3} a subset of {1, 2, 3} because all of the elements in {1, 2, 2, 3} are contained in {1, 2, 3}? However, {1, 2, 2, 3} isn't part of the power set of {1, 2, 3}, right? Thanks!
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75 views

Determine if a lottery system is profitable

I have a problem where I'm supposed to determine if a lottery system is profitable. I solved the problem and found it to be profitable, but I am not 100% sure about all of my calculations. Below is ...
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109 views

Creating random numbers matching mean and standard deviation

I know how to compute mean and standard deviation for a given probe. But how do I the opposite? Given is the wanted mean and standard deviation and I want to create the probes. In other words: What ...
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122 views

Number of expected runs in a fair coin flip

I have this question in mind Say we flip a particular fair coin for 20 times and note down the sequences. Then what is the number of expected runs we can get? Thought process: Let X = number of runs ...
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743 views

Let X = the time between two successive arrivals at the drive-up window of a local bank…

Let $X$ = the time between two successive arrivals at the drive-up window of a local bank. $X$ has an exponential distribution with $\lambda = 2$. That is the probability density of $X$ is $f(X | \...
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If X~beta($\theta$,1) then Z=-ln(x) ~exp(1/$\theta$)

I keep getting this, $P(Z\leq z)=P(-ln(X) \leq z) = P(X \leq e^{-z}) = \int_{0}^{e^{-z}} \theta t^{\theta-1}dt = t^{\theta}|_{0}^{e^{-z}}=e^{-z\theta}$ $\frac{d}{dz}[e^{-z\theta}]=-\theta e^{-z\...
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$F(t)=(\beta/\alpha)\int_{0}^{t}\frac{(t/\alpha)^{\beta-1}}{[1+(t/\alpha)^{\beta}]^2}dt$

Compute $$F(t)=(\beta/\alpha)\int_{0}^{t}\frac{(t/\alpha)^{\beta-1}}{[1+(t/\alpha)^{\beta}]^2}dt$$ My Attempt : Let $u=(t/\alpha)\Rightarrow \alpha du=dt$ so, $$F(t)=(\beta/\alpha)\int_{0}^{(t/\...
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149 views

Normal approximation of binomial distribution with finite n corrections

I know I can approximate binomial distribution $B(n,p)$ with normal distribution $N(np,np(1-p))$. For finite $n$, I assume there are correction terms for mean and variance of the normal distribution,...
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To calculate the expectation of a variable with a given pdf

Let the pdf of a random variable X be given by $f(x)=ae^{-x^2-bx}, -\infty<x<\infty$. If $E(X)=-\frac{1}{2}$, then (A)$a=\frac{1}{\sqrt{\pi}}e^{-1/4},b=1$ (B)$a=\frac{1}{\sqrt{\pi}}e^{-1/4},b=-...
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80 views

Find the expectations of the largest and smallest order statistics $X_{(n)}$ and $X_{(1)}$ respectively. Uniform distribution

Suppose that $X_1,\cdots, X_n$ are independent random variables from uniform distribution on interval $(\theta_1,\theta_2)$, $\theta_2>\theta_1>0$. It is know that $T(X)=(X_{(1)},X_{(n)})$ is ...
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Poisson time arrival process

Assume that calls arrive at a call center according to a Poisson arrival process with a rate of λ calls per hour. For $0 <= s < t$, what is the probability of $N((0,s]) = m$, when conditioned on ...
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51 views

Why is my Binomial Theorem not working?

Ok this might seem like a dumb question which I think it is but it is really bugging me. Lets say you have a 1 in 5 chance at something ( doesn't matter what) and you get 2 attempts at it. By basic ...
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37 views

Some clarifications and a question on basic probability.

I have a few questions and some clarifications. CLARIFICATIONS: 1. Assume we roll 2 four sided dice. What is P({sum of the rolls is even})? I answered the question correctly I: Odd + Odd = Even J: ...
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106 views

Central limit theorem: What is the probability than more than 36 randomly chosen songs are required to fill a program which is 76 minutes long?

I am a little stumped by what this question is asking. A large playlist consists of songs with times which have mean 2 minutes ten seconds and standard deviation 15 seconds. What is the probability ...
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41 views

Find MLE and show that it is unbiased.

I'm trying to solve a problem but not sure how to approach it because of the weird density function: Would appreciate any constructive advice!
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35 views

Do we have these kinds of “mean value”?

Let $x_1,\ldots,x_n$ be positive numbers. We want to find a number $x$ such that the sum $$ \sum_{i=1}^n|x-x_i| $$ get its minimal value. I know that such $x$ may be not unique. Nevertheless do we ...
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43 views

Probability of [X<a + bZ, X<Y, X>Z]?

I'm trying to derive the following probability: $\Pr \left[ {X < a + bZ,X < Y,X > Z} \right]$ where X,Y, and Z follow exponential distribution with parameters $\lambda_x$, $\lambda_y$, and $...
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421 views

How to prove Laplace distribution is scale mixture of Gaussians?? [closed]

How does one prove the Laplace distribution is a scale mixture of Gaussians? I.e, how does one show that $X \sim \text{Laplace}(\lambda)$ is a scale mixture of Normal $Y \sim N(0,\tau)$ and ...
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142 views

Properties of weighted average

Consider a set of real positive numbers $\{x_1,x_2,...,x_n\}$ all greater than a positive real number $s$. Consider $n$ positive weights $w_i$ s.t. $\sum_{i=1}^{n}w_i=1$. Show that $\frac{1}{n}\sum_{i=...
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70 views

Combination or Probability Word Problem

Harrt and his 2 friends decided to visit a a famous wizard restaurant. In how many ways can Harry pay for their bill of 1000 pesos if he has 6 one thousand peso bills and, 4 five hundred peso bills, ...
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24 views

Algebra Simplification (Statistics)

I am working through a proof in a statistics course, and I am having trouble following one equality. How is the following equality true? $$n\left[\sum X_i^2-\frac{\left(\sum X_i\right)^2}{n}\right]=...
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22 views

Probability problem for at least two events?

Assume that the probability that you find a typo on a given page in any book is approximately $0.0025$. Find an approximation for the probability that in then next $1000$ pages that you read, you will ...
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64 views

How to find the probability distribution function from the following Distribution function?

How to find the corresponding probability distribution function for the following distribution function ? $$F (x)= \left\{ \begin{array}{ll} 0 & \text{if } x<0 \\ x^2 & \text{if } 0\...
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58 views

Moment generating function from a $f(x)$

Given a function, I am required to find the $E[X]$ and $Var[X]$ without using MGF and then have to verify the answer by finding the MGF. I found the $E[X]$ and MGF. However, I am unable to verify ...
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133 views

Can you help me with this Markov Chain question?

The Problem: Prove that if the number of States in a Markov Chain is M, and that state j can be reached from state i, then it can be reached in M steps or less. The work: I assumed by contradiction ...
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Why does $P[X \leq p_{k/100}] = \frac{k}{100}$ work?

I am a student studying an introductory course in statistics. We have recently covered percentiles and the following equation was given to us without explanation: $$P\left[X \leq p_{\frac{k}{100}}\...
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34 views

Variance of the compound sum

Trying to solve a Variance evaluation problem: Now I'm not sure how to evaluate those two terms on the right hand side of the last equality... Would appreciate any help.
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55 views

How many pairs of positive constants $a, b$ exist such that $P(a < X < b) = 0.95$, where $X$ has a chi-squared distribution?

$X$ is a chi-squared distribution with $n$ degrees of freedom (sum of the squares of $n$ $N(0,1)$ variables). How many pairs of positive constants $a,b$ exist such that $$ P(a < X < b) = 0.95 $$...
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48 views

For a compound random variable $\Sigma^N_{i=1} X_i$. find $Cov(N, S)$

For a compound random variable $\Sigma^N_{i=1} X_i$. find $Cov(N, S)$ I know $Cov(X,Y) = E[SN] - E[S]E[N]$ but i'm not sure how to find andy of these values.
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60 views

How to estimate mean and variance of a normal distribution given the numbers?

Given the numbers generated in a normal distribution: $5.3299, 4.2537, 3.1502, 3.7032, 1.6070, 6.3923, 3.1181, 6.5941, 3.5281, 4.7433, 0.1077, 1.5977, 5.4920, 1.7220, 4.1547, 2.2799$ How would I ...
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559 views

Why does randomness exhibit a pattern in the long run?

!!! Layman here so please avoid complex math and answers. Random (usually pseudorandom) events are usually characterized along these lines: Each outcome in a trial experiment must be i.i.d.; i.e. ...
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208 views

Probability of woman receiving positive mammogram and having cancer

The probability that a randomly selected US woman will have breast cancer in their lifetime is 0.12. Women over 40 are advised to have regular mammograms because early detection of breast cancer means ...
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75 views

Conditional Probability of A given B, is it not just A?

If Conditional Probably is defined as $P(A\mid B) = \frac{\displaystyle P(A \cap B)}{\displaystyle P(B)}$, and $P(A \cap B)$ is defined as $P(A) \times P(B)$, is $P(A \mid B) = P(A)$?
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59 views

Interesting probability rule for predicting the outcome of a trial

I remember a friend of mine claiming something a bit funny. It went along the lines of: Given that one has observed $n$ trials of an experiment and $s$ of them have been successes (as opposed to ...
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51 views

how to create unbiased estimator in uniform distribution

$X_{1}, X_{2},...X_{n},n\geqslant 2, $ is a random sample from unif[$\theta -1, \theta +1$] Followed with the problem, I got T(X)=($X_{(1)}, X_{(n)} $) is sufficient but not complete, But I got ...
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105 views

Question about sum of chi-squared distribution

I want to prove that the sum of two independent chi-squared random variables is a chi-squared random variable. I am supposed to only use the fact that if $Q$ has a chi-squared distribution with ...
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35 views

Show this converges in distribution to 0

Let $\{ X_n:n \geq 1 \}$ such that $$f_{X_n} = \begin{cases} (n-1)/2 &\mbox{if } -1/n <x<1/n \\ 1/n & \mbox{if } n<x<n+1 \end{cases}$$ Show that this converges to $0$ in ...
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44 views

Minimize the following function

I'm trying to find the value of $n_1$ which minimize the following function: $$Var(\overline{X}_{Str,n})=0.25\left(\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{20-n_1}\right )$$ I tried to solve the ...