# Linked Questions

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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 $... • 2,540 0 votes 0 answers 71 views ### Making a t-distribution with sample size less than 30 [duplicate] Our teacher said that we can use the graph of a standard normal distribution whenever the sample size is greater than or equal to 30 because of CLT. How can we make a graph of a$t$-distribution if ... • 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 ... • 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 $$\frac{\Gamma \left(\frac{k+1}{2}\right)}{\Gamma\left(\frac{k}{2}\... • 535 30 votes 2 answers 24k views ### Why is a T distribution used for hypothesis testing a linear regression coefficient? In practice, using a standard T-test to check the significance of a linear regression coefficient is common practice. The mechanics of the calculation make sense to me. Why is it that the T-... • 403 12 votes 2 answers 5k views ### How can I obtain a Cauchy distribution from two standard normal distributions? I am interested in Let X\sim N(0,1), Y \sim N(0,1) independently. Show \frac{X}{X+Y} is a Cauchy random variable. My work: f_{X,Y}(x,y)=\frac{1}{2\pi} e^{\frac{-1}{2}(x^2+y^2)}, -\infty&... • 2,169 19 votes 1 answer 11k views ### Why Test Statistic for the Pearson Correlation Coefficient is \frac {r\sqrt{n-2}}{\sqrt{1-r^2}} I am learning hypothesis testing for Pearson Correlation Coefficient. The source did not explain why the test statistic$$\frac {r\sqrt{n-2}}{\sqrt{1-r^2}}$$satisfy T distribution with n-2 degree ... • 37k 3 votes 1 answer 3k views ### What is the difference between auxiliary variable and Latent variable? What is the difference between auxiliary variable and Latent variable when we talk about joint posterior density (when it is in complex form). Notice that, my work is based on continuous random ... • 139 3 votes 1 answer 492 views ### Understanding Why (or Why Not) a T-Test Require Normally Distributed Data? [duplicate] This is a concept that I have always struggled to understand: We can write the formula for a Two Sampled T- Test (https://en.wikipedia.org/wiki/Student%27s_t-test) to compare the sample averages from ... 2 votes 0 answers 1k views ### A Normal random variable divided by a Chi-squared random variable I am looking to find the pdf of the ratio Z = \frac{X}{Y}, where X \sim Gaussian and Y \sim Chi-squared are indepedent random variables. A reference is good enough if you do not want to write ... • 21 3 votes 0 answers 1k views ### Chi-square origin of the name What are the origins of the names and letters in these distributions: What is the origin of the name in chi-square distribution \chi_k^2? And the origin of t in student's t-test? And naming ... • 187 5 votes 1 answer 258 views ### Deegrees of freedom of Student's distribution I'm trying to figure out the distribution of this statistic:$$S=\frac{\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}}{\sqrt{\hat{\sigma}^2/\sigma^2}}$$Where: \overline{X}=\frac{1}{n} \sum_{i=1}^n ... • 301 1 vote 1 answer 453 views ### How to prove (\hat{X}-\mu)/(\hat{S}/\sqrt{n}) is student t with n-1 degrees of freedom if X_i are iid N(\mu, \sigma)? It is commonly stated that if X_i are iid N(\mu, \sigma), then with \hat{X} the sample mean, and \hat{S} the sample error (sample standard deviation), then \frac{ \hat{X}-\mu}{\hat{S}/\sqrt{n}... 0 votes 1 answer 442 views ### convergence in distribution: show limit is standard normal (Y_n)_{n\geq 1} is iid standard normal. Then how can I prove the following:$$\frac{Y_1}{(n^{-1}\sum_{k=1}^n Y_k^2)^{1/2}}\rightarrow N(0,1)$$• 29 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 thatW = Z_1^2 + Z_2^2 ...
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