Linked Questions

2 votes
0 answers
321 views

Derive t-distribution PDF kernel from its definition [duplicate]

In the Wikipedia article on the t-distribution (https://en.wikipedia.org/wiki/Student%27s_t-distribution), they define the random variable - $$Y = \frac{(\bar{X} - \mu)}{\frac{S}{\sqrt n}}$$ Where $...
ryu576's user avatar
  • 2,540
0 votes
0 answers
71 views

Making a t-distribution with sample size less than 30 [duplicate]

Our teacher said that we can use the graph of a standard normal distribution whenever the sample size is greater than or equal to 30 because of CLT. How can we make a graph of a $t$-distribution if ...
soupless's user avatar
  • 101
0 votes
0 answers
27 views

In simple linear regression model, why do we calculate the confidence interval for slope parameter using t-distribution? [duplicate]

I'm taking a regression analysis course and we were studying simple linear regression. I've understood how slope $$ \hat\beta_1 follows \space N(0,\sigma^2 / S_{xx}) $$ and is normally disributed. And ...
aroma's user avatar
  • 123
35 votes
3 answers
18k views

Student t as mixture of gaussian

Using the student t-distribution with $k > 0$ degrees of freedom, location parameter $l$ and scale parameter $s$ having density $$\frac{\Gamma \left(\frac{k+1}{2}\right)}{\Gamma\left(\frac{k}{2}\...
Salih Ucan's user avatar
30 votes
2 answers
24k views

Why is a T distribution used for hypothesis testing a linear regression coefficient?

In practice, using a standard T-test to check the significance of a linear regression coefficient is common practice. The mechanics of the calculation make sense to me. Why is it that the T-...
Nate Parke's user avatar
12 votes
2 answers
5k views

How can I obtain a Cauchy distribution from two standard normal distributions?

I am interested in Let $X\sim N(0,1), Y \sim N(0,1)$ independently. Show $\frac{X}{X+Y}$ is a Cauchy random variable. My work: $f_{X,Y}(x,y)=\frac{1}{2\pi} e^{\frac{-1}{2}(x^2+y^2)}, -\infty&...
Ron Snow's user avatar
  • 2,169
19 votes
1 answer
11k views

Why Test Statistic for the Pearson Correlation Coefficient is $\frac {r\sqrt{n-2}}{\sqrt{1-r^2}}$

I am learning hypothesis testing for Pearson Correlation Coefficient. The source did not explain why the test statistic $$\frac {r\sqrt{n-2}}{\sqrt{1-r^2}}$$ satisfy T distribution with $n-2$ degree ...
Haitao Du's user avatar
  • 37k
3 votes
1 answer
3k views

What is the difference between auxiliary variable and Latent variable?

What is the difference between auxiliary variable and Latent variable when we talk about joint posterior density (when it is in complex form). Notice that, my work is based on continuous random ...
Manal's user avatar
  • 139
3 votes
1 answer
492 views

Understanding Why (or Why Not) a T-Test Require Normally Distributed Data? [duplicate]

This is a concept that I have always struggled to understand: We can write the formula for a Two Sampled T- Test (https://en.wikipedia.org/wiki/Student%27s_t-test) to compare the sample averages from ...
stats_noob's user avatar
2 votes
0 answers
1k views

A Normal random variable divided by a Chi-squared random variable

I am looking to find the pdf of the ratio $Z = \frac{X}{Y}$, where $X \sim$ Gaussian and $Y \sim$ Chi-squared are indepedent random variables. A reference is good enough if you do not want to write ...
GAA's user avatar
  • 21
3 votes
0 answers
1k views

Chi-square origin of the name

What are the origins of the names and letters in these distributions: What is the origin of the name in chi-square distribution $\chi_k^2$? And the origin of $t$ in student's $t$-test? And naming $...
user2925716's user avatar
5 votes
1 answer
258 views

Deegrees of freedom of Student's distribution

I'm trying to figure out the distribution of this statistic: $$S=\frac{\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}}{\sqrt{\hat{\sigma}^2/\sigma^2}}$$ Where: $\overline{X}=\frac{1}{n} \sum_{i=1}^n ...
ChicagoCubs's user avatar
1 vote
1 answer
453 views

How to prove $(\hat{X}-\mu)/(\hat{S}/\sqrt{n})$ is student t with $n-1$ degrees of freedom if $X_i$ are iid $N(\mu, \sigma)$?

It is commonly stated that if $X_i$ are iid $N(\mu, \sigma)$, then with $\hat{X}$ the sample mean, and $\hat{S}$ the sample error (sample standard deviation), then $\frac{ \hat{X}-\mu}{\hat{S}/\sqrt{n}...
travelingbones's user avatar
0 votes
1 answer
442 views

convergence in distribution: show limit is standard normal

$(Y_n)_{n\geq 1}$ is iid standard normal. Then how can I prove the following: $$\frac{Y_1}{(n^{-1}\sum_{k=1}^n Y_k^2)^{1/2}}\rightarrow N(0,1)$$
xxxxxxxx's user avatar
3 votes
1 answer
158 views

Distribution of $\frac{Z_1^2 + Z_2^2}{Z_1+Z_2}$ where $Z_1, Z_2$ are standard normals

Let $Z_1, Z_2 \sim \mathcal{N}(0,1)$ be i.i.d random variables. I wish to find the distribution of \begin{align} \frac{Z_1^2 + Z_2^2}{Z_1+Z_2} \,. \end{align} It is well known that $W = Z_1^2 + Z_2^2 ...
lemmykc's user avatar
  • 91

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