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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A joint PDF question: $\displaystyle f(x,y)=1-\frac x3-\frac y3$

Really stuck on this problem: If $\displaystyle f(x,y)=1-\frac x3-\frac y3$ for $0 \le x \le 2$ and $0 \le y \le h,$ the find $h$. I know I need to integrate but confused how to set it up. ...
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12 views

Conditional Expectation, Orthogonality, and Correlation

I know that if $\epsilon$ and $x$ are independent, then $E[\epsilon|x]=E[\epsilon]$ and Cov$(\epsilon,x)=0$. However, $E[\epsilon|x]=E[\epsilon]=0$ implies Cov$(\epsilon,x)=0$ iff $\epsilon$ and $x$ ...
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2answers
36 views

probability, expectation, variance

A 10-digit long number is picked randomly and each digit's pick is independent and has an equal probability of being picked (1/9 because there's digits 1 to 9). Let $X = \#\{\text{missing digits}\}$ ...
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1answer
26 views

Expectation: E(X-Y) and Variance: Var(X-Y) in statistics

There are $n$ students and you pick a student replacement so each pick is independent . $p$ is the portion of the student body that will vote for A instead of B for student council president. Let $X ...
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22 views

Average of Cumulative Distributions

I have numerous cumulative distributions of the normal distribution function that all test the against the same mean and standard deviation, but different test values (x). My objective is to obtain an ...
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1answer
60 views

Probability of event in normal distribution

Let $X$ be a random variable that is normally distributed and $X_1,\ldots,X_n$ be (independet) copies of $X$, then we can estimate this probability by using a simple Monte-Carlo estimator: $p := P (X ...
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0answers
37 views

Finding the nth moment of the geometric distribution: Why does interchanging the derivative and summation operators not work after n=1?

I am trying (and failing) to find a recursive formula for the $nth$ moment of a geometric distribution. I have arrived at bogus results, and I think it has something to do with the convergence of the ...
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0answers
22 views

Covariance of two random variable from vector if density function is given

How can I find $Cov(X_1,X_2)$ if I have density function $f(x_1,x_2,x_3)$? I suppose that $Cov(X_1,X_2) = E(X_1 X_2) - EX_1EX_2$. Because I have density function of random vector thus I can find ...
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36 views

Urgent Find the CDF of U [duplicate]

I am having problems with Part 2. I know the upper limit of Y is x+u in the formula. But what about the limits of X ? Please help me !!
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If I have mean, median, kurtosis, and skew, can I derive a skewed normal cumulative distribution function?

I'm thinking I'd have to transform the values using either natural logarithm, or estimating lambda and performing a box cot transformation. however... I can easily derive skew and lambda and all ...
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1answer
28 views

Find marginal and conditional distributions [closed]

Consider the probabiility density function $f_{X_1, X_2}(x_1, x_2) = \left\{\begin{matrix}\frac{1}{8x_2} \exp\left\{ -\left( \frac{x_1}{2x_2} + \frac{x_2}{4}\right)\right\}, & x_1 > 0, ...
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1answer
35 views

Confusion in Posterior Probability Calculation

I know posterior probability as, $P(\theta|x)= [(P(x|\theta)*(P(\theta))/(P(x))]$, as given in http://en.wikipedia.org/wiki/Posterior_probability I am slightly confused with the term ...
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1answer
44 views

Prove (n choose k)(p)^(k)(1-p)^(n-k) = 1

I am taking a statistics course and found this question in my textbook. I am trying to incorporate the binomial theorem into the proof, but am hitting a wall. Any suggestions would be greatly ...
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0answers
17 views

steps in factor analysis in matlab

i know basically how to express PCA mathematicaly,but i would like to know steps that should be used for factor analysis,let us consider some matrix,for instance ...
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1answer
22 views

Excel version of Skewed Normal Cumulative Distribution Function

I'm trying to deskew some distributions based on checks that verify Skew and Kurtosis and within range, if not, I was going to attempt to deskew the data. On wikipedia, there's a mathematical ...
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0answers
9 views

Is it acceptable to manually adjust cyclic index for weekend days when creating a forecast for each day of the week?

I'm creating a call volume forecast for an inbound call center. I'm using the exploration forecast method, and all works well except for weekend days. I've found that if I apply a common mulitplier ...
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2answers
27 views

Finding out missing observation $a$ and $b$

The mean of $a,b,8,5,10$ is $6$ and the variance is $6.8$. find out value of $a$ and $b$ I tried something like this: $$\bar{x}=6=\frac{a+b+8+5+10}{5}$$ $$7=a+b$$ Then i have put the value in ...
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68 views

Concentration inequality of weighted sum of random variables given a tail inequality

I tried to solve the exercise below which can be seen as a generalization of the Bernstein's concentration inequality. However, I have difficulty bounding the moment generating function of $Z$ (see ...
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1answer
9 views

Finding out number of observation

There are $n$ scores $X_1,X_2,X_3,....,X_n$ and their sum is $80$ and sum of their squares is $400$ then which among them is the probable value of $n$ A)$10$ B)$9$ C)$15$ ...
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1answer
27 views

Shapiro-Wilk test

I am trying to determine if a given sample comes from a Normal distribution. For that purpose I want to perform a Shapiro-Wilk test in the way stated on wikipedia. My concern comes with the vector ...
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0answers
15 views

Cumulative Distribution Function of skewed data

how can I derive (in excel) a cumulative distribution function of a set of values, when I assume I want to correct for all skews? I have derived a skew as well as kurtosis. I am working with this ...
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1answer
30 views

Difficulty in understanding statistics and inference

I have a random variable, X, with cdf $F$ and pdf $f$. I want to estimate a parameter of $F$, say mean, $\mu$. So what do I do? I construct an estimator, $Y_n$, with several random variable X1, ...
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1answer
33 views

Probability question? [duplicate]

A manager is interviewing 3 applicants for a job. The duration of each interview follows an exponential distribution with parameter 1/2, time being measured in hours. The interviews are scheduled to ...
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1answer
15 views

Hypothesis Testing Using Summations

Assuming that n=12 and $\sum_{i=1}^n Y_i^{2}\tag{} = 42 $ perform a level $\alpha=0.05 $ hypothesis test of the following hypothesis. $$H_o=\theta=\theta_o=2$$ against the alternative $$H_a: \theta ...
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2answers
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Is the variation of the mean 0, and the variance of standard deviation just the variance?

Just trying to get some properties straight. Was looking at a problem online which reads: Let X be a random variable with mean μ and variance σ2. What is the variance of X/σ+10μ? The solution of ...
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2answers
27 views

clarify basic probability question

You have a loaded die. $A =$ even numbers $B =$ odd numbers $P(A)=3/4$ $P(B)=1/4$ $P(\{1\})=1/12$ $P(\{2\})=1/4$ How was $P(\{1\})$ and $P(\{2\})$ calculated? Is it beacuse there are 3 odds ...
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29 views

Introduction to statistics

Let $X$ be uniformly distributed on $[a,b]$. Find a function $f$ such that $Y=f(X)$ is exponentially distributed with parameter $1$. How do I do this?
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1answer
36 views

If $X$ has Unif[0,2] distribution, then what is the density of $Y=X1_{(0,1)}(X)$

If $X$ has Unif[0,2] distribution, then what is the density of $Y=X1_{(0,1)}(X)$? (1 is the indicator function) What I have is that $Y=0$ with probability $P(1\le X\le 2)=F_X(2)-F_X(1)=\frac{1}{2}$ ...
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2answers
27 views

Binomial/Negative Binomial Distribution? Why not Poisson here?

When I looked at the below problem, I thought of Poisson immediately. I converted the rate to making 9/10 shots. However the answer told me to use the binomial/negative binomial distribution for parts ...
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20 views

Finding significant peaks

I have a small doubt. My real data looks like this $Y$ values are random values of integers from $0$ to $2000$. $X$ values run like $1,2,3,4,5,\ldots$ to $2$ million. Now, my task is to identify ...
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1answer
84 views

Probability of waiting for an interview

I have the following problem, for which I am having trouble finding a solution: A manager is interviewing 3 applicants for a job. The duration of each interview follows an exponential distribution ...
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2answers
31 views

Joint distribution: show the components of the joint distribution are independent.

Very odd question I think... Show that if $(X,Y)$ is a random vector in $\mathbb{R}^{2}$ with density $f_{(X,Y)}(x,y) = f(x)g(y)$ for a pair of non-negative functions $f$ and $g$, then $X$ has ...
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1answer
27 views

Finding the Probability of A=2B or 2A=B?

Three Companies are bidding on a contract ... etc , Company (A) is twice as good as company (B), And Company (B) is three times as good as Company (C). What's the probability of each company winning ...
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43 views

Poisson/ jump process distribution for process $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$

For the process: $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$, where $X(t)$ is a poisson process with paramater $\lambda$, and: $J_k$ are i.i.d . random variables (jumps). $B(t)$=brownian motion. I want to ...
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3answers
54 views

Random Variable Problems?

Can someone show me how to work this out? I can't get the answers in the boxes.
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2answers
93 views

How to calculate historical data knowing only yearly percentage increase and data for year 2015?

How do I calculate historical yearly market worth knowing only its yearly growth rate and its martket worth for year 2015? Example: According to MarketLine, the global home improvement market is ...
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1answer
27 views

If F is a cumulative distribution function, then $\lim_{x\to\infty}F(x)=1$

Let $F$ be a c.d.f for a random variable X and let $$x_1<x_2<\cdots<x_n<x_{n+1}<\cdots$$ be a monotonic increasing sequence that converges to $+\infty$. I've managed to do a tight proof ...
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1answer
18 views

Probability that a sample comes from one of two distributions

Let's say I have two normal distributions with means $\mu_1$, $\mu_2$ and standard deviations $\sigma_1$, $\sigma_2$ (which I know). I am handed a random variate from one of the distributions (I don't ...
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0answers
26 views

Minimizing variance of difference of two means in one-way ANOVA

From Kutner et al, fifth edition, p. 772: Refer to Rehabilitation therapy Problem 16.9. The sample sizes for below-average, average, and above-average physical fitness groups are to be $n$, $kn$, ...
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3answers
48 views

Simplify function with polynomial via least-squares

I want to "adjust" (simplify) $f(x)$, a function, by $g(x)$, a polynomial, via least-squares. I want to write code for that. Apperently my code is issuing wrong results, so I was wondering if my ...
2
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1answer
76 views

-ln(0.1) equalling to ln(10)?

I am having quite a headache wrapping my head around this solution. I do not understand the first line where they get lambda = ln(10) from statement to the left. Somebody please explain this to me. ...
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3answers
30 views

Determine the Cumulative Distributive Distribution(CDF) of a truncated value?

It is the last part(part h) that I am having problems with. I know you use integration and then split it into 2 parts. But how exactly do you do it ? A detailed answer would be very helpful ! ...
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1answer
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Quick Bayes Question

I drew a tree diagram but I can't get the right answer listed below. Can anyone show me how to do this? The New York State Health Department reports a 10% rate of the HIV virus for the “at-risk” ...
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1answer
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Mathematical expectation of variable X

X is random variable (see the image) Can E[X] get more simplified Thank you Vladimir, I was on the similar trace:
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2answers
24 views

Is there a statistical test to use when comparing sample means without knowledge of population or sample standard deviation?

The issue I am hoping to solve: In Google Analytics (the web analytics platform), one can compare, lets say, the pages/visit for two traffic sources. As an example: Facebook: 300 visits, 6.2 ...
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1answer
36 views

Finding an efficient estimator for $\theta$ in $U[0, \theta]$ in terms of the sample maximum

This question appeared in a past exam paper, in the form: Let $X = (X_1\dotsc X_n)\in\mathbb{R}^n$ be an i.i.d. sample from $U[0, \theta], \theta>0$ Apply Rao-Blackwell's theorem to the unbiased ...
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probabilty/statistics game - What makes the highest possibility?

I just came across this game today at a market where you can bet 5, 10, or 20 kroner(danish valuta) on 6 different colors and then a cup with a coloured dice and a normal dice gets flipped after you ...
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23 views

Find the Probability of at Least 2 students , Probability of three are given .

The Question : The Probability of success of three students X,Y and Z in one examination are 1/5 , 1/4 and 1/3 respectively , Find the probability of success of at least two. Choices : (A. 1/6) ...
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28 views

Why does an exponential random graph model belong to the exponential family?

The exponential random graph model is defined as, $$ P_\theta(Y=y)=\frac{\exp\{\theta^ts(y)\}}{c(\theta)}.$$ Where $y \in \mathcal{Y}$ the set of all possible networks, $\theta = ...
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1answer
22 views

Calculating Inflection Point in Real World Data

I can think of a few ways to get a good grasp of this problem from an experimental perspective, but I'm looking for a more general explanation that can be applied to different situations. If this ...