Introduction to Continuous and Mixed Random Variables

In probability and statistics, random variables are classified into two major types: continuous and discrete. Continuous random variables can take an infinite number of possible values, while mixed random variables can incorporate both continuous and discrete characteristics. This blog will explore the basic concepts of continuous random variables, probability density functions, and special distributions such as the uniform and normal distributions.

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1.1 Continuous Random Variables

A continuous random variable can take any value within a given interval, often representing measurements like time, height, or temperature. The value of a continuous random variable is not countable but can be any real number within a specific range.

1.1.1 Continuous Random Variables and Their Distributions

For continuous random variables, the Probability Density Function (PDF) describes the likelihood of the random variable taking a value within a specific interval. The PDF is a function $f(x)$ such that:

• $f(x)\ge 0$ for all $x$,
• The total area under the curve of the PDF is equal to 1:

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=1$

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1.1.2 Probability Density Function

The PDF provides the probability that a continuous random variable falls within a given range. The probability that the random variable $X$ falls between $a$ and $b$ is given by the integral of the PDF over that interval:

$P(a\le X\le b)={\int }_a^{b}f(x)dx$

For example, if $f(x)$ is the PDF of the variable $X$, and we want to calculate the probability that $X$ is between 1 and 3, we would compute:

$P(1\le X\le 3)={\int }_1^{3}f(x)dx$

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1.1.3 Expected Value and Variance

The expected value $E(X)$ of a continuous random variable is a measure of its central tendency and is calculated as:

$E(X)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}xf(x)dx$

The variance $Var(X)$ measures the spread or dispersion of the random variable and is given by:

$Var(X)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x–E(X){)}^{2}f(x)dx$

The expected value gives us the mean or average of the random variable, while the variance provides a measure of how much the values of the variable deviate from the mean.

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1.1.4 Functions of Continuous Random Variables

Sometimes, we are interested in the behavior of a function of a continuous random variable. If $Y=g(X)$, where $X$ is a continuous random variable, we can find the distribution of $Y$ using the change of variables formula. This formula transforms the PDF of $X$ into the PDF of $Y$ by adjusting for the relationship between the variables.

For example, if $Y=2X$, the PDF of $Y$ will differ from that of $X$, and we need to account for this transformation using the change of variables formula.

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1.1.5 Solved Problems

To apply the concepts discussed, let’s consider a solved problem:

Problem: Given a random variable $X$ that follows a normal distribution with mean $\mu =50$ and standard deviation $\sigma =5$, calculate the probability that $X$ is between 45 and 55.

Solution: Using the PDF of the normal distribution:

$P(45\le X\le 55)={\int }_{45}^{55}\frac{1}{\sqrt{2\pi }5}{e}^{-\frac{(x–50{)}^2}{2\cdot 5^2}}dx$

The result can be calculated using standard techniques or a numerical integration tool to find the probability.

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1.2 Special Distributions

Various special distributions are widely used in probability theory, each with unique properties.

1.2.1 Uniform Distribution

A uniform distribution is one where all outcomes within a specific interval are equally likely. The PDF for a uniform distribution between $a$ and $b$ is given by:

$f(x)=\frac{1}{b–a}$ for $a\le x\le b$

For example, the probability of drawing a number between 2 and 5 from a uniform distribution between 1 and 6 is:

$P(2\le X\le 5)={\int }_2^5\frac{1}{6–1}dx=\frac{3}{5}=0.6$

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1.2.2 Exponential Distribution

The exponential distribution is commonly used to model the time between independent events occurring at a constant rate. The PDF is:

$f(x)=\lambda e^{-\lambda x}$, for $x\ge 0$

Where $\lambda$ is the rate parameter. The mean of the exponential distribution is $\frac{1}{\lambda }$, and the variance is $\frac{1}{{\lambda }^2}$.

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1.2.3 Normal (Gaussian) Distribution

The normal distribution is one of the most commonly used distributions in statistics. It is defined by two parameters: the mean $\mu$ and the standard deviation $\sigma$. The PDF of the normal distribution is:

$f(x)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{(x–\mu {)}^2}{2{\sigma }^2}}$

The normal distribution is symmetric around the mean, and many natural phenomena, such as heights and test scores, are approximately normally distributed.

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1.2.4 Gamma Distribution

The gamma distribution is used in various applications, including waiting time distributions in queuing theory. The PDF of the gamma distribution is:

$f(x;k,\theta )=\frac{1}{{\theta }^k\mathrm{\Gamma }(k)}{x}^{k-1}{e}^{-x/\theta }$

Where $k$ is the shape parameter, $\theta$ is the scale parameter, and $\mathrm{\Gamma }(k)$ is the gamma function.

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1.2.5 Other Distributions

Other important continuous distributions include the beta distribution, chi-square distribution, and log-normal distribution. Each of these has unique properties and applications.

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1.2.6 Solved Problems

Problem: Find the probability that a random variable $X$ following a uniform distribution between 0 and 10 takes a value between 3 and 7.

Solution: Using the uniform distribution formula:

$P(3\le X\le 7)={\int }_3^7\frac{1}{10–0}dx=\frac{4}{10}=0.4$

Thus, the probability is 0.4, or 40%.

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1.3 Mixed Random Variables

A mixed random variable is a combination of both discrete and continuous random variables. This type of variable arises in many practical scenarios, such as insurance claims where a certain probability mass may be assigned to zero, but the remaining claims follow a continuous distribution.

1.3.1 Mixed Random Variables

Mixed random variables contain both discrete and continuous components. For example, let’s consider an insurance payout, where there’s a certain probability that no payout is made (a discrete value of 0), and a continuous range of possible payouts when a claim is made.

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1.3.2 Using the Delta Function

In probability theory, the delta function $\delta (x)$ is used to handle mixed random variables. The delta function allows us to represent a discrete component within a continuous distribution. It’s often used in scenarios where there’s a spike or point mass at a particular value.

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1.3.3 Solved Problems

Problem: Suppose an insurance company offers a policy where 20% of customers have no claims (i.e., $X=0$) and the claims for the remaining 80% follow an exponential distribution with $\lambda =0.1$. Calculate the expected payout.

Solution:

The expected value is calculated as:

$E(X)=0\times 0.2+{\int }_0^{\mathrm{\infty }}x\cdot 0.1e^{-0.1x}dx$

This integral evaluates to $10$, so the expected payout is $10$.

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Conclusion

In this blog, we explored the basic concepts of continuous and mixed random variables, including their probability distributions, expected values, and special distributions like the uniform and normal distributions. Understanding these topics is crucial for analyzing real-world scenarios in fields ranging from engineering to finance.

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