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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.

Statistics is the science of the collection, organization, and interpretation of data. It deals with many aspects of data, which includes the planning of data collection in terms of the design of surveys and experiments. (From Wikipedia)

More specifically, 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 mathematical analysis. (From Wikipedia)

For questions which are more generally about collecting and treating data, it is advised that you post your question on Cross Validated and DSSE.

37109 questions
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Lottery, cumulative distribution function, variance

Suppose a lottery is played like this: You must pay $\$5$ to play. Then, you select three numbers from $\{0, 1, 2, ..., 9\}$, with each of the three numbers being different (order does not matter). Let’s suppose that you choose the numbers $4, 7$,…
Natalie
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Let X be a discrete random variable with expected value E(X)...

Let $X$ be a discrete random variable with expected value $E(X)$. Further, suppose there is a $1/4$ probability of $X$ being exactly $2$ units away from $E(X)$, a $1/4$ probability of $X$ being exactly $3$ units away from $E(X)$, and a $1/2$…
Natalie
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Prove the computational formula of Anderson-Darling test statistic.

The Anderson-Darling test statistic is defined as $$n\int_{-\infty}^\infty \frac{(F_n(x) - F(x))^2}{F(x)(1 - F(x))}dF(x)$$ and there is a computational formula $$A^2 = -n - S$$ where $$S = \sum_{k=1}^n\frac{2k-1}{n}\left(\ln F(Y_k) + \ln(1 -…
Bowen
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standard deviation of x= 121 divided by 121

$\{X_1, X2, \ldots, X_{121}\}$ are independent and identically distributed random variables such that $E(X_i)= 3$ and $\mathrm{Var}(X_i)= 25$. What is the standard deviation of their average? In other words, what is the standard deviation of $\bar…
snowdoniastudio
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How to tell if there is equal variance in a box plot?

I'm trying to decide if the variance in these groups in this boxplot are equal, so how can I tell how much variation each group has just looking at the box plot? And how can I tell if they all have equal variance? Here is the boxplot:
jn025
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Limits of integration for random variable

Suppose you have two random variables $X$ and $Y$. If $X \sim N(0,1)$, $Y \sim N(0,1)$ and you want to find k s.t. $\mathbb P(X+Y >k)=0.01$, how would you do this? I am having a hard time finding the limits of integration. How would you generalize…
lord12
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calculating $P(X_n=\max(X_1,X_2,\ldots, X_n))$

Suppose $X_1,X_2,\ldots X_{n-1}\sim U(0,1)$, $X_n\sim \exp(\frac{1}{2})$. Else, suppose $X_1,X_2\ldots, X_n$ are independent. How can I calculate $P(X_n=\max(X_1,X_2\ldots, X_n))$
user2135285
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Proving variance of U-statistics is decreasing

I read Wassily Hoeffding's paper "a class of statistics with asymptotically normal distribution". In proving "$n\sigma^{2}(U_{n})$ is decreasing in n" in Theorem 5.2, it simply says "using (5.33) and (5.31)". Yet this is not obvious to me. Is there…
Jie Wei
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Determining how well a curve fits data

What are some commonly used ways to examine how well a curve fits a given set of data? I am aware of the R Squared test but I was wondering if there are other tests that take into account the appearance of the curves as well. To be more precise, I…
Sandeep Silwal
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Power of a statistical test

I was working on the following problem: Consider two probability density functions on $[0,1]: f_0(x) = 1$, and $f_1(x) = 2x$. Among all tests of the null hypothesis $H_0: X \sim f_0(x)$ versus the alternative $X \sim f_1(x)$, with significance level…
user151850
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How to interpret right hand side of Cumulative Distribution Function

What is the type (?) of this term: $X \leq x$ in the definition of the CDF? $F_{X}(x) = P(X \leq x)$ Is $X \leq x$ a set? Is it real valued? I know that ${P}$ is a function that maps to $[0,1]$. Does it always map a set to $[0,1]$? How would I read…
Bryan Glazer
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Compute variance of "tree" random variable

Let A be a random variable defined as: With probability $p[i]$, the random variable $B[i]$ is drawn $B[i] ~ N[mu[i],sigma[i]]$ probabilities $p[i]$ sum up to one I know how to compute the mean, which is given by: $$E[A] = p[1]*mu[1] + .. + …
Breugem
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Generate Sequence with given variance and mean

Is there a simple way to do this by hand? I mostly want to do this for quick "straight face" testing to see if something works, so ideally I want (for example) a sequence with mean 1 and variance 2 with as many 0's as possible (for ease of…
soandos
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What are some of the disadvantages of working with log likelihood function instead of likelihood function?

As the title suggested, I want to know why people often use log likelihood function, instead of likelihood function by itself. What I know, is that, if $\hat{\theta}$ is the maximum of a likelihood function $f(\theta; \mathbf{x})$, then it is also…
Minke
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Question from Fundamentals of Biostatistics

Suppose we observe 84 alchoholics with cirrhosis of the liver, of whom 29 have hepatomas, that is, a liver-cell carcinoma. Suppose we know, based on a large sample, that the risk of hepatoma among alcoholics without cirrhosis of the liver is…
Dillweed
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