Continuous Random Variables

Let $S$$\large S$ be the sample space associated with some experiment $E$$\large E$. A random variable $X$$\large X$ is a function that assigns a real number $X(s)$$\large X(s)$ to each elementary event $s\in S$$\large s \in S$. In this case, as the outcome of the experiment, i.e. the specific $s$$\large s$, is not predetermined, then the value of $X(s)$$\large X(s)$ is not fixed. This means that the value of the random variable is determined by the specific outcome of the experiment.

A continuous random variable can take an uncountably infinite number of values.

Probability Density Function

For a continuous random variable $X$$\large X$, the probability density function $f$$\large f$ is a non-negative function defined for all real $x\in (-\mathrm{\infty },\mathrm{\infty })$$\large x \in (-\infty, \infty)$, such that, for any set $A$$\large A$:

Properties

• $P\{X\in (-\mathrm{\infty },\mathrm{\infty })\}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)dx=1$$\large P\{X \in (-\infty, \infty)\} = \int\limits_{-\infty}^{\infty}f(x)dx = 1$

Cumulative Distribution Function

For a continuous random variable $X$$\large X$, the cumulative distribution function $F(a)$$\large F(a)$ is defined as:

Properties

• $a<b⟹F(a)\le F(b)$$\large a \lt b \implies \Large F(a) \le F(b)$
• $\underset{a\to \mathrm{\infty }}{lim}F(a)=1$$\large \lim\limits_{a \to \infty} F(a) = 1$
• $\underset{a\to -\mathrm{\infty }}{lim}F(a)=0$$\large \lim\limits_{a \to -\infty} F(a) = 0$
• $\underset{n\to \mathrm{\infty }}{lim}F(a_n)=F(a)$$\large \lim\limits_{n \to \infty} F(a_n) = F(a)$ for any decreasing sequence $a_n$$\Large a_n$ that converges to $a$$\large a$
• $P\{a<X\le b\}=F(b)-F(a)$$\large P\{a \lt X \le b\} = F(b) - F(a)$

Expected Value

For a continuous random variable $X$$\large X$, the expected value $E[X]$$\large E[X]$ is defined as:

Expected Value of a Function of a Continuous Random Variable

For a real-valued function $Y=g(X)$$\large Y = g(X)$ of a random variable $X$$\large X$, the expected value $E[Y]$$\large E[Y]$ is defined as:

Thus we can show:

where $a$$\large a$ and $b$$\large b$ are constants.

Variance

For a continuous random variable $X$$\large X$, the variance $Var(X)$$\large Var(X)$ is defined as:

where $\mu =E[X]$$\large \mu = E[X]$.

Thus we can show:

where $a$$\large a$ and $b$$\large b$ are constants.

Standard Deviation

For a continuous random variable $X$$\large X$, the standard deviation $SD(X)$$\large SD(X)$ is defined as:

Uniform Random Variable

A uniform random variable $X$$\large X$ is a continuous random variable that is uniform in the interval $(\alpha ,\beta )$$\large (\alpha, \beta)$, and its probability density function is given by:

The cumulative density function $F(a)$$\large F(a)$ is defined as:

Properties

• $E[X]=\frac{\alpha +\beta }{2}$$\large E[X] = \frac{\alpha + \beta}{2}$
• $Var(X)=\frac{(\beta -\alpha {)}^2}{12}$$\large Var(X) = \frac{(\beta - \alpha)^2}{12}$
• $SD(X)=\frac{\beta -\alpha }{2\sqrt{3}}$$\large SD(X) = \frac{\beta - \alpha}{2\sqrt{3}}$

Normal Random Variable

A normal random variable $X$$\large X$ is a continuous random variable that is normally distributed with parameters $\mu$$\large \mu$ (mean) and $\sigma$$\large \sigma$ (standard deviation), denoted by $N(\mu ,{\sigma }^2),$$\large N(\mu, \sigma^2),$ with its probability density function is given by:

Properties

• $E[X]=\mu$$\large E[X] = \mu$
• $Var(X)={\sigma }^2$$\large Var(X) = \sigma^2$
• $SD(X)=\sigma$$\large SD(X) = \sigma$
• $X:N(\mu ,{\sigma }^2)\text{and}Y=aX+b⟹Y:N(a\mu +b,a^2{\sigma }^2)$$\large X: N(\mu, \sigma^2) \: \text{and} \: Y = aX + b \implies Y : N(a\mu + b, a^2\sigma^2)$

Standard Normal Random Variable

A standard normal random variable $X$$\large X$ is a normal random variable with parameters $\mu =0$$\large \mu = 0$ and $\sigma =1$$\large \sigma = 1$, and its probability density function is defined as:

The cumulative distribution function $\mathrm{\Phi }(x)$$\large \Phi(x)$ of a standard normal random variable is defined by:

Properties

• $X:N(\mu ,{\sigma }^2)⟹F_X(a)=\mathrm{\Phi }(\frac{a-\mu }{\sigma })$$\large X : N(\mu, \sigma^2) \implies F_X(a) = \Phi(\frac{a - \mu}{\sigma})$

Chi-Square Random Variable

Given $\nu$$\large \nu$ standard random variables $Z_1,Z_2,...,Z_{\nu }$$\large Z_1, Z_2, ..., Z_\nu$, the chi-square random variable $X$$\large X$ is given by:

where $\nu$$\large \nu$ represents degrees of freedom.

The probability density function of a chi-square random variable is the chi-square distribution, and is defined by:

where $\mathrm{\Gamma }(\alpha )$$\large \Gamma(\alpha)$ is the gamma function and is defined as:

The cumulative distribution function $F(x)$$\large F(x)$ of the chi-square distribution is defined by:

where $\gamma (x,\alpha )$$\large \gamma(x, \alpha)$ is the incomplete gamma function and is defined as:

Properties

• $E[X]=\nu$$\large E[X] = \nu$
• $Var(X)=2\nu$$\large Var(X) = 2 \nu$
• $SD(X)=\sqrt{2\nu }$$\large SD(X) = \sqrt{2 \nu}$

T-Statistic

Given a normal random variable $Z$$\large Z$ and a $\chi$$\large \chi$-squared random variable $U$$\large U$, the T-statistic $T$$\large T$ is a random variable with parameters $\mu$$\large \mu$ (population mean of $Z$$\large Z$), $\overline{x}$$\large \bar{x}$ (sample mean), $\sigma$$\large \sigma$ (population standard deviation of $Z$$\large Z$), $s$$\large s$ (sample standard deviation) and $n$$\large n$ (degrees of freedom of $\chi$$\large \chi$), and is defined as:

The probability distribution function of the T-statistic is the Student's $t$$\large t$-distribution, and is defined by:

where $\nu$$\large \nu$ is the degrees of freedom ($\nu =n-1$$\large \nu = n - 1$) and $\mathrm{\Gamma }(\alpha )$$\large \Gamma(\alpha)$ is the gamma function.

The cumulative distribution function $F(t)$$\large F(t)$ of the T-statistic is defined by:

Properties

• $E[T]=0$$\large E[T] = 0$, for $\nu >1$$\large \nu \gt 1$
• $Var(T)=\frac{\nu }{\nu -2}$$\large Var(T) = \frac{\nu}{\nu - 2}$, for $\nu >2$$\large \nu \gt 2$
• $SD(T)=\sqrt{\frac{\nu }{\nu -2}}$$\large SD(T) = \sqrt{\frac{\nu}{\nu - 2}}$, for $\nu >2$$\large \nu \gt 2$

F-Statistic

Given chi-squared random variables $X$$\large X$ and $Y$$\large Y$, with degrees of freedom ${\nu }_1$$\large \nu_1$ and ${\nu }_2$$\large \nu_2$ respectively, the F-statistic $W$$\large W$ is defined as:

The probability distribution function $f(W)$$\large f(W)$ is the Snedecor's $F$$\large F$-distribution, and is defined by:

The cumulative distribution function $F(W)$$\large F(W)$ of the F-statistic is defined by:

where $k=\frac{{\nu }_2}{{\nu }_2+{\nu }_{1}W}$$\large k = \frac{\nu_2}{\nu_2 + \nu_1W}$ and $I_k$$\large I_k$ is the incomplete beta function,

where $B$$\large B$ is the beta function,

Properties

• $E[W]=\frac{{\nu }_2}{{\nu }_2-2}$$\large E[W] = \frac{\nu_2}{\nu_2 - 2}$, for ${\nu }_2>2$$\large \nu_2 \gt 2$
• $Var(T)=\frac{2{\nu }_2^2({\nu }_1+{\nu }_2-2)}{{\nu }_1({\nu }_2-2{)}^2({\nu }_2-4)}$$\large Var(T) = \frac{2\nu_2^2(\nu_1 + \nu_2 - 2)}{\nu_1(\nu_2 - 2)^2 (\nu_2 - 4)}$, for ${\nu }_2>4$$\large \nu_2 \gt 4$
• $SD(T)=\sqrt{\frac{2{\nu }_2^2({\nu }_1+{\nu }_2-2)}{{\nu }_1({\nu }_2-2{)}^2({\nu }_2-4)}}$$\large SD(T) = \sqrt{\frac{2\nu_2^2(\nu_1 + \nu_2 - 2)}{\nu_1(\nu_2 - 2)^2 (\nu_2 - 4)}}$, for ${\nu }_2>4$$\large \nu_2 \gt 4$

Exponential Random Variable

An exponential random variable $X$$\large X$ is a continuous random variable with parameter $\lambda$$\large \lambda$ with its probability density function, the exponential distribution defined as:

The cumulative distribution function $F(a)$$\large F(a)$ is defined by:

Properties

• $E[X]=\frac{1}{\lambda }$$\large E[X] = \frac{1}{\lambda}$
• $Var(X)=\frac{1}{{\lambda }^2}$$\large Var(X) = \frac{1}{\lambda^2}$
• $SD(X)=\frac{1}{\lambda }$$\large SD(X) = \frac{1}{\lambda}$

Logistic Random Variable

A logistic random variable $X$$\large X$ is a continuous random variable with mean parameter $\mu$$\large \mu$ and scale parameter $s$$\large s$, with its probability density function, the logistic distribution defined as:

The cumulative distribution function is defined by:

Properties

• $E[X]=\mu$$\large E[X] = \mu$
• $Var(X)=\frac{s^2{\pi }^2}{3}$$\large Var(X) = \frac{s^2 \pi^2}{3}$
• $S(X)=\frac{s\pi }{\sqrt{3}}$$\large S(X) = \frac{s \pi}{\sqrt{3}}$

Gamma Random Variable

A gamma random variable $X$$\large X$ is a continuous random variable with parameters $\lambda >0$$\large \lambda \gt 0$ and $\alpha >0$$\large \alpha \gt 0$, with the gamma distribution as the probability density function, defined as:

where $\mathrm{\Gamma }(\alpha )$$\large \Gamma(\alpha)$ is the gamma function.

Properties

• $E[X]=\frac{\alpha }{\lambda }$$\large E[X] = \frac{\alpha}{\lambda}$
• $Var(X)=\frac{\alpha }{{\lambda }^2}$$\large Var(X) = \frac{\alpha}{\lambda^2}$
• $SD(X)=\frac{\sqrt{\alpha }}{\lambda }$$\large SD(X) = \frac{\sqrt{\alpha}}{\lambda}$

Memoryless Random Variable

A memoryless random variable $X$$\large X$ is one that satisfies the followings condition:

An exponential random variable is memoryless.