The t-distribution was discovered by W. S. Gosset and R.A. Fisher. The entries in Student’s t table entries are the critical values (percentiles) for the t distribution. The applications of Student’s t distribution are related to (i) the sampling distribution of the mean $\overline{x}$, (ii) the distribution of a difference $(\overline{x}_1 – \overline{x}_2)$ of two independent populations, (iii) the distribution of two paired (dependent) populations, and (iv) the significance of correlation coefficient. It is also used for constructing confidence intervals for small samples. The Student’s t distribution is a crucial tool in statistical analysis, especially when dealing with small sample sizes. It helps us make informed decisions based on our data, even when the population standard deviation is unknown.

The Student’s t variable can be generated by dividing the standard normal random variable ($Z$) with the square root of a $\chi^2_{v}$ random variable. The $\chi^2_v$ is itself divided by its parameter $v$. That is

\begin{align*}
t_v &= \frac{x – \mu }{s_v} = \frac{\tfrac{(x-\mu)}{\sigma} }{\sqrt{\dfrac{\frac{v\times s^2_v}{\sigma^2} } {v}}}\\
&= \frac{Z}{\sqrt{\dfrac{\chi^2_v}{v}}}
\end{align*}

where

### PDF of Student’s t Distribution

The PDF of t having $v$ degrees of freedom is

$$p(t_v) = K_v (1+\frac{t^2}{v})^{-\frac{v+1}{2}}$$

where

$$K_v = \frac{\Gamma \left[ \frac{(v+1}{2} \right]}{\sqrt{v\pi} \left(\frac{v}{2}\right) }$$

The t distribution is symmetric about zero and wider than normal density. It has one mode and it tends to be normal as $v\rightarrow \infty$. Note that $\Gamma(x)$ indicates the Gamma function.

### Moments of t Distribution

Since the t distribution is symmetric and its PDF is centered at zero, the expectation (average), the median, and the mode are all zero for the t distribution with $v$ degrees of freedom. The variance ($\sigma^2$) equals $\frac{v}{v-2}$ and kurtosis is $\frac{6}{v-4}$.

For bivariate normal population, the distribution of correlation coefficient $r$ is linked with Student’s t distribution through transformation:

$$\frac{r}{\sqrt{\frac{1-r^2}{n-2}}}\rightarrow t_{n-2}$$

### Generation of Pseudo Random t Variates

The following algorithm can be used to generate random variates from Student’s $t(v)$ distribution using serially generated independent uniform $U(0,1)$ random variates. For example,

Let $n=v$ (the degrees of freedom)

$C = -2n$

Repeat
$t = 2 \times U(0, 1) – 1$
$u = 2 \times U(0, 1) – 1$
$r = t^2 + u^2$
Until
$r < 1$
Return
$t \times \sqrt{\frac{n \times (r^C – 1)}{r}}$

### Student’s t Table

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## Chi-Square Distribution ($\chi^2$) Made Easy

The Chi-square distribution is a continuous probability distribution that is used in many hypothesis tests. The Chi-Square statistic always results in a positive value.

A Chi-Square variate (with $v$ degrees of freedom (df)) is the sum of $v$ independent, squared standard normal variates ($\sum\limits_{i=1}^v z_i^2$). It is denoted by $\chi^2_v$. The variance $s^2$ from a sample of normally distributed observations is distributed as $\chi^2$ with $v$ (the df) as a parameter referred to as df of the calculated variance. Symbolically,

$$\frac{v\cdot s^2}{\sigma^2} \sim \chi^2_v$$

The variance $s^2$ for $n$ observations from a $N(\mu, \sigma^2)$, the df is equal to $v=n-1$. The Chi-Square distribution is also used for the contingency (analysis of frequency) tables as an approximation to the distribution of complex statistics. All the families of Chi-Square distribution are specified by their degrees of freedom.

### Chi-Square Distribution Case of the Gamma Distribution

The Chi-Square distribution is a particular case of the Gamma Distribution, the pdf is

$$P_{\chi^2}(x) = [2^{v/2}\Gamma(v/2)]^{-1} \chi^{(v-2)/2}e^{-x/2}, \quad x\ge 0$$

where $\Gamma(x)$ is the Gamma Distribution.

### Normal Approximation to $\chi^2$

Method 1: The PDF and df of Chi-Square can be approximated by the normal distribution. For large $v$ df, the first two moments $z=\frac{(X-v)}{\sqrt{2v}}$, $X\sim \chi^2$.

Method 2: Fisher approximation (compensates the skewness of $X$)

$$\sqrt{2X} – \sqrt{2v-1} \sim N(0, 1)$$

Method 3: Approximation by Wilson and Hilferty is quite accurate. Defining $A=\frac{2}{9v}$, we have

$$\frac{\sqrt[3]{(X/v)}-1+A}{\sqrt{A}}\sim N(0, 1)$$

For the determination of percentage points

$$\chi^2_{v[P]}=v[z_P\sqrt{A}+1-A]^3$$

### Generating Pseudo Random Variates

Following the schema allows the generation of random variates from $\chi^2_v$ distribution with $v>2$ df. It requires to generate serially random variates from the standard uniform $U(0,1)$ distribution.

Let $n=v$ degrees of freedom

\begin{align*}
C1 &= 1 + \sqrt{2/e} \approx 1.8577638850\\
C2 &= \sqrt{n/2}\\
C3 &= \frac{3n^2-2}{3n(n-2)}\\
C4 &= \frac{4}{n-2}\\
C5 &= n-2\\
\end{align*}

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## Standard Normal Table (2012)

A standard normal table, also called the unit normal table or Z-table, is a table for the values of Φ calculated mathematically, and these are the values from the cumulative normal distribution function. A standard normal distribution table is used to find the probability that a statistic is observed below, above, or between values on the standard normal distribution, and by extension, any normal distribution. Since probability tables cannot be printed for every normal distribution, as there is an infinite variety (families) of normal distributions, it is common practice to convert a normal to a standard normal and then use the standard normal table to find the required probabilities (area under the normal curve).

The standard normal curve is symmetrical, so the table can be used for values going in any direction, for example, a negative 0.45 or positive 0.45 has an area of 0.1736.

The Standard Normal distribution is used in various hypothesis testing procedures such as tests on single means, the difference between two means, and tests on proportions. The Standard Normal distribution has a mean of 0 and a standard deviation of 1.

The values inside the given table represent the areas under the standard normal curve for values between 0 and the relative z-score.

The table value for Z is 1 minus the value of the cumulative normal distribution.

### Standard Normal Table (Area Under the Normal Curve)

For example, the value for 1.96 is $P(Z>1.96) = 0.0250$.

### Standard Normal Table (Summary)

• A table of values for the cumulative distribution function (CDF) of the standard normal distribution.
• The standard normal distribution has a mean of 0 and a standard deviation of 1.
• This table shows the probability that a standard normal variable will be less than a certain value (z-score).

For further details see Standard Normal

See about the measure of asymmetry

Probability in R Language