Questions tagged [statistics]
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.
2,334
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Intuition behind using complementary CDF to compute expectation for nonnegative random variables
I've read the proof for why $\int_0^\infty P(X >x)dx=E[X]$ for nonnegative random variables (located here) and understand its mechanics, but I'm having trouble understanding the intuition behind ...
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vote
2
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Inclusion–exclusion principle; what is $(-1)^{n+1}$
could somebody kindly confirm that my understanding of inclusion-exclusion matches it's formula.
for a 3 sets example; we add 3 unions, subtract the total of all 3 pairwise intersections and add the ...
- 261
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Density of sum of two independent uniform random variables on $[0,1]$
I am trying to understand an example from my textbook.
Let's say $Z = X + Y$, where $X$ and $Y$ are independent uniform random variables with range $[0,1]$. Then the PDF
is
$$f(z) = \begin{cases}
z &...
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2
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Proof of upper-tail inequality for standard normal distribution
$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$,
$$
\mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>.
$$
user754
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answers
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Motivation behind standard deviation?
Let's take the numbers 0-10. Their mean is 5, and the individual deviations from 5 are
-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
And so the average (magnitude of) ...
100
votes
11
answers
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What's so special about standard deviation?
Equivalently, about variance?
I realize it measures the spread of a distribution, but many other metrics could do the same (e.g., the average absolute deviation). What is its deeper significance? ...
- 1,345
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votes
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answers
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Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of continuous X
Let $X$ be a random variable with a continuous and strictly increasing c.d.f. $F$ (so that the quantile function $F^{−1}$ is well-defined). Define a new random variable $Y$ by $Y = F(X)$. Show that $Y$ ...
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Recommend a statistics fundamentals book
To give you some background, I have a grasp on the basics of statistics and probability theory and even remember touching Bayes theorem at the university data mining course. But being a few years away ...
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5
answers
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Proof of $\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}$
It's a standard result that given $X_1,\cdots ,X_n $ random sample from $N(\mu,\sigma^2)$, the random variable $$\frac{(n-1)S^2}{\sigma^2}$$ has a chi-square distribution with $(n-1)$ degrees of ...
- 8,419
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Unbiased Estimator for a Uniform Variable Support
Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \left[ 0, \theta \right] $. Now I need to estimate $ \theta $ based on $N$ observations and I want the estimator to be ...
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112
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Variance of sample variance?
What is the variance of the sample variance? In other words I am looking for $\mathrm{Var}(S^2)$.
I have started by expanding out $\mathrm{Var}(S^2)$ into $E(S^4) - [E(S^2)]^2$
I know that $[E(S^2)]^...
- 1,329
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votes
7
answers
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Intuitive Explanation of Bessel's Correction
When calculating a sample variance a factor of $N-1$ appears instead of $N$ (see this link ). Does anybody have an intuitive way of explaining this to students who need to use this fact but maybe ...
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Finding UMVUE of $\theta$ when the underlying distribution is exponential distribution
Hi I'm solving some exercise problems in my text : "A Course in Mathematical Statistics".
I'm in the chapter "Point estimation" now, and I want to find a UMVUE of $\theta$ where $X_1 ,...,X_n$ are i....
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3
answers
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X,Y are independent exponentially distributed then what is the distribution of X/(X+Y)
Been crushing my head with this exercise. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything together.
The exercise goes as ...
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How do you differentiate the likelihood function for the uniform distribution in finding the M.L.E.?
There is a classic problem:
Suppose that $X_1,\ldots,X_n$ form an i.i.d. sample from a uniform distribution on the interval $(0,\theta)$, where $\theta>0$ is unknown. I would like to find the MLE ...
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