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Results tagged with statistics
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user 791458
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.
1
vote
a best critical region of two parameters in normal distribution
You can manipulate your expression finding the following critical region
$$\Sigma_i \left(X_i+\frac{1}{3}\right)^2 \geq k^*$$
Now, under $H_0$, the distribution is known: it's a noncentral chi-squared …
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2
votes
Accepted
Cramer-Rao lower bound for exponential distribution
Yes you did.
the lower bound for unbiased estimators of $\lambda$ is $V(T)\geq\frac{\lambda^2}{n}$
Using Lehmann-Scheffé Lemma you can find the UMVUE estimator of $\lambda$
$\hat{\lambda}=\frac{n-1 …
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0
votes
Location-scale family
Exponential distribution does not belong to a location - scale family. Actually it belongs to a "scale family"
Definition
A scale family of distributions has densities of the form
$$ \bbox[5px,border …
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0
votes
Bayesian Reliability
the likelihood is the following
$$p(\mathbf{x}|\theta)\propto \theta^3(1-\theta)^8$$
which is a Binomial model
What does it suggest to you?
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0
votes
All cumulative distribution function follows a U[0,1]
This theorem is named "integral transformation theorem "
As you can see after few passages you get
$$F_Y(y)=F_X[F_X^{-1}(y)]$$
now it is evident that applying both $F$ and $F^{-1}$ to any $y$ they can …
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1
vote
Accepted
normal distribution candy problem
The probability to mark the candies exactly of 5 older kids is the probability to chose 5 older kids among the 35 total, say
$$\frac{\binom{15}{5}\binom{20}{0}}{\binom{35}{5}}$$
(an hypergeometric...) …
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1
vote
Accepted
Conditional Probability with Poisson's variable
Note that
$$P(S|S>0)=\frac{1}{1-e^{-0.3}}\frac{e^{-0.3}0.3^s}{s!}$$
Thus
$$P(S=1|S>0)=\frac{0.3}{e^{0.3}-1}\approx 85.75\%$$
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0
votes
Accepted
what is the distribution for a chi-squired variable divided by its degrees of freedom?
No, it does not. I do not know why you think so.
As you know
$$Y\sim \chi_{(m)}^2=Gamma\Bigg(\frac{m}{2};\frac{1}{2}\Bigg)$$
Thus
$$\frac{1}{m}Y\sim Gamma\Bigg(\frac{m}{2};\frac{m}{2}\Bigg)$$
The proo …
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0
votes
What is the first moment of ML estimator theta=n/sum(xi)
I do not know if your really have to calculate this expectation....but reading the title, if you are only interested in detecting if your MLE is unbiased, the answer is the following
By Jensen's inequ …
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0
votes
Calculating Minimum Exam Score to Pass a Course
so I need to score 15% or higher on the exam? Or am I missing something?
Your answer is not "totally" to waste.
first you did not take 25% in the project score but you took $0.3\times0.85=25.5\%$
…
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1
vote
Accepted
Cramér-Rao Lower Bound-Exponential distribution
-$n$ random sample you get
$$V(T)\geq \frac{k^2\theta^{2k}}{n}$$
b) Observe that $T=\overline{X}_n$ is unbiased estimator for $\theta$ and it is a function of $S=\Sigma_iX_i$, complete and sufficient statistics … can be useful to calculate the expectation of
$$T_k=[\Sigma_iX_i]^k$$
This is easy because $\Sigma_iX_i$ has a known distribution (it's a gamma) and $T_k$ is function of $S$, complete and sufficient statistics …
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0
votes
Accepted
Statistics problem about computing UMVUE(Uniformly Minimum Variance Unbiased Estimator)
There are several ways to show that $T=\overline{X}_n$ is the UMVUE for $p$
This is one; I chose this because it allow us to do further considerations about that important estimator:
$$\mathbb{E}[\ove …
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2
votes
Accepted
Determine the values of $\mu$ and $\sigma^2$.
You know that
$$P(X\leq 4)=0.8413$$
Knowing also that $E(X^2)-\mu^2=\sigma^2$ you get
$$P\left(Z \leq\frac{4-\mu}{\sqrt{10-\mu^2} } \right)=0.8413$$
Using the tables you get
$$\frac{4-\mu}{\sqrt{10-\m …
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2
votes
Accepted
Positive parameters for Gamma random variables
The gamma density is the following, for $x>0$
$$f_X(x,a,b)=\frac{b^a}{\Gamma(a)}x^{a-1}e^{-bx}$$
it is easy to prove that its integral cannot converge if $a,b$ are not both positive
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1
vote
Need help understanding the motivation for parametric statistical models
This made me wonder what the motivation for parametric models is; I mean why do we want to infer a parameter instead of a measure?
Additional question: How would one extend the two definitions such …
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1
vote
Accepted
What is the expected value of the estimator?
Any estimator is a function of (only of) the data. thus tipically
$$\hat{\theta}=t(\mathbf{x})$$
and thus, by definition,
$$\mathbb{E}[\hat{\theta}]=\int_T tf(t)dt$$
Example...
given a simple random …
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1
vote
Is it possible for some results to be "more" statistically significant than others provided ...
Would it be correct to call the results from the first pair of sets "more" statistically significant since the p value is lower?
Yes, absolutely right. The p-value represents a measure of significan …
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2
votes
Accepted
Finding the mle of a log normal distribution
The question is very very simple.
But the important thing you have to keep also in mind is that the likelihood is defined unless a moltipilicative constant.
Thus if you look at your density that is
$$ …
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3
votes
Accepted
Proving that ${-1 \leq \operatorname{Corr}(X,Y) \leq 1}$
Let's consider the rv $Z=Y+aX$ and let's calculate its variance.
$$V(Z)=V(Y)+a^2V(X)+2a cov(X,Y)\geq0$$
As per the fact that variance cannot be negative, the above expression is a 2nd degree inequalit …
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2
votes
Accepted
Test if population standard deviation is less than 7.2.
First you have to assume Normality...
Then your Hypothesis' System to be verified is the following
$$\begin{cases}
\mathcal{H}_0: & \sigma^2=51.84 \\
\mathcal{H}_1: & \sigma^2<51.84
\end{cases}$$
th …
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3
votes
Accepted
meaning of II symbol in a math equation
It is a Capital i, as "indicator function"
you can also find $\mathbb{1}$ and indicates that the function is 1 for the values in the braces and zero elsewhere
Example,
Suppose you have the following …
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1
vote
System of three equations, one term squared
An easier way to solve the problem is to realize that
$$\mathbb{E}(X^2)=\mathbb{V}(X)+\mathbb{E}^2(X)$$
and $c=1-0.625-a$
thus setting the following system
$$\begin{cases}
-5a+0.28125b+4\cdot(0.375-a) …
- 32.9k
1
vote
Accepted
Find a complete sufficient statistic T for the interior of a triangle
So far so good.
To find the density of $\max(X+Y)$ first derive the distribution of $Z=X+Y$
$$F_Z(z)=\frac{z^2}{2}\cdot\frac{2}{\theta^2}=\frac{z^2}{\theta^2}$$
Thus
$$f_Z(z)=\frac{2 z}{\theta^2}\cdo …
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2
votes
Accepted
Standard Normal Table
I suppose you are wondering to have the probability of the desired quantile, that is
$$P\left(Z\leq -\frac{1}{3}\right)$$
Often Standard normal tables are tabulated only for the positive quantiles, bu …
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3
votes
Accepted
An exact and an approximate confidence interval for a Poisson distribution
Let's start with the approximate CI:
For $n$ greater enough ($n=10$ is borderline but actually enough to get your CI with a Gaussian distribution) you can apply CLT in the following way
$$\frac{\overl …
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2
votes
Accepted
Find UMVU estimators for an exponential distribution
As you surely know, Negative Exponential distribution belongs to the Exponential Family thus $S=\Sigma_i X_i$ is CSS (Complete, minimal and Sufficient Statistic).
The UMVUE of $\theta$ has been discus …
- 32.9k
1
vote
Accepted
plug-in estimator of variance
It is true because
$$\mathbb{E}[\overline{X}_n-\mu]^2=\mathbb{V}[\overline{X}_n]=\mathbb{V}\Bigg[\frac{1}{n}\Sigma_i X_i\Bigg]=\frac{1}{n^2}\mathbb{V}[\Sigma_i X_i]=\frac{1}{n^2} n\sigma^2=\frac{\sigm …
- 32.9k
1
vote
Question about the Z
Later on, when I come to Z-score test of a significant level, the
formula Z becomes $Z=\frac{X-\mu}{\sigma}\sqrt{n}$
This formula is incorrect. You find
$$Z=\frac{\overline{X}_n-\mu}{\sigma}\sqrt{n} …
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0
votes
Accepted
UMVU Estimator of Bernoulli Distribution
Using Rao - Blackwell & Lehmann - Scheffé together, the UMVUE is
$$\mathbb{E}[T|S]$$
Where $T$ is an unbiased estimator for $p$ and $S$ is a Complete and Sufficient Statistic.
Let's set $ T=X_1$ and …
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1
vote
Variance of a single random variable with two terms.
As noted your density is not exactly a pdf. Anyway it is clearly a mixture of 2 different negative exp pdf
$$f_X(x)=a f_1(x)+(1-a)f_2(x)$$
and obviously
$$f_1(x)=0.5e^{-0.5 x}$$
and
$$f_2(x)=0.25 e^{- …
- 32.9k
1
vote
Need to find sample distribution with population distribution p given and sample size n given
For i) the given answer is correct and it is simply due to the fact that
$$\mathbb{E}[\overline{X}_n]=\mu=40\%$$
For ii), using the continuity correction factor, you get
$$\mathbb{P}[\Sigma_i X_i>50]= …
- 32.9k
0
votes
Need some help with STATS
a)
$$\mathbb{P}[700<X<850]=\Phi\Bigg[\frac{850-800}{70}\Bigg]-\Phi\Bigg[\frac{700-800}{70}\Bigg]=$$
$$=\Phi[0.714]-\Phi[-1.429]=0.762-0.077=0.686\approx 68.6\%$$
b), c) can you proceed by yourself? th …
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0
votes
What's the correct null and alternative hypothesis?
Mercury levels higher than two parts per billion in drinking water are unsafe.
Reading the statement, the obvious system of hypothesis to be verified is the following
$$\begin{cases}
\mathcal{H}_0, …
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2
votes
Accepted
What does the term "unbiased estimator" mean?
$S^2$ is unbiased estimator for the population variance $\sigma^2$ because, as per definition
$$\mathbb{E}[S^2]=\sigma^2$$
there are other and most important properties of an estimator, i.e. consisten …
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0
votes
Accepted
statistics random sample question
telling the truth...the correct answer should be
D1)
$$1-\Phi(2.125)$$
but I figure they want you to say D)
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1
vote
Accepted
consistency of MLE estimator example
If you are not familiar with SLLN, there are several other ways to approach the problem,
The simplest: a property of ML Estimators is that they are consistent.
Consistency you have to prove is $\ha …
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1
vote
Infinite bias for MLE
Hint and some notes:
The problem here is that the Bias you are searching is this:
$\mathbb{E}[\frac{nk}{X}-N]$
The difficulty here is to calculate $\mathbb{E}[\frac{1}{X}]$
Without doing a lot of …
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1
vote
Show $\frac{(n-1)S^2}{\theta}$ is pivotal quantity of random sample $Y_1,...,Y_n$ from $N(\t...
It is correct. Find the 2 quantiles of the chi-squared assuming equiprobable tails and solve w.r.t. $\theta$
Your pivotal qty is not the only one possible.
Check the definition of $S^2$ because the on …
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2
votes
Is a mean of 12 unusual for a sample of size 30 from Exp(1/10)?
No, it is not unusual.
"Usually" do define an "unusual" value one refers to the fact that this events happens in less than $5\%$ of the times.
In your exercise, before getting the distribution of the …
- 32.9k
1
vote
Finding $E[X]$ of a probability distribution
Your expression is neither a pdf nor a CDF but it should be a pmf, a probability mass function of a geometric rv. As noted, the exponent should be
$k$, if $k \in \{0;1;2;\dots\}$
or
$k-1$, if $k \in \ …
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6
votes
Accepted
Why Geometric distribution has different expectations but the same variance?
Because the two distributions are linked in the following way
$$Y=X-1$$
Thus using expectation property you get that
$$E(Y)=E(X)-1$$
That is
$$E(Y)=\frac{1}{p}-1=\frac{1-p}{p}$$
But, using variance pr …
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4
votes
Accepted
Need help with Pearson criterion (chi-square) to check if the distribution is uniform
If your distribution was uniform you should have had $156/3=52$ observations each value. Now apply chi-squared test
$$\chi^2=\frac{(51-52)^2}{52}+\frac{(40-52)^2}{52}+\frac{(65-52)^2}{52}$$
and then c …
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1
vote
Accepted
Calculating the minimum size of a sample
Your answer is correct.
Thus the minimum size is $n=107$. Perhaps there is an error in the book's solution
For b) set the sample mean $\overline{X}=\frac{70}{107}$ and use the confidence intervals for …
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1
vote
Why use a Z test rather than T test for confidence interval of a population proportion?
in few words:
T test is used to estimate the mean when the population distribution is known as Gaussian but wit unknown variance
the test on propotion is a test on a mean of a bernulli population. U …
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0
votes
Accepted
Which statistics test to be used in the following test?
There are several ways to solve the problem
Z test for proportions
chi square test (non parametric test)
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0
votes
Statistics z score mean, sd and variance
Start with $X\sim N(\mu;\sigma^2)$. This means that X is normally distributed with mean $\mu$ and standard deviation $\sigma$
If you transform you rv in the following way
$$Z=\frac{X-\mu}{\sigma}$$
yo …
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1
vote
Accepted
How can it be derived? (Law of the unconscious statistician)
The quick answer is that $Y=e^X\sim \text{Lognormal}$ thus its mean is well known
If you want to do all the calculation with the gaussian distribution, it is not difficult; try, it is a good exercise
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2
votes
Accepted
Method of Moments estimation
I did not do all the calculations because it is only a matter to solve algebraic systems but I explain you how to do...
To calculate MoM's estimators, the first thing you have to do is to express your …
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1
vote
Help me calculate the probability and the related questions.
First question: if Bob's probability is the half of the other 49 this means that
$$49\times 2p+p=1$$
$$p=\frac{1}{99}$$
Now you can proceed with the second question
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1
vote
Accepted
Expected absolute value of the difference between a random variable and its mean
Your expression is also a measure of dispersion. In fact it is a distance between the data points and $\mu$. It is not optimal becasuse (it is easy to prove) that the minimum distance
$$\int_{-\infty} …
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