In probability and statistics, a transformation of a random variable involves applying a function to that variable. This is a fundamental concept, especially in actuarial exams, as it allows us to derive the probability distribution of a new random variable based on the known distribution of an original one. Understanding transformations is crucial for modeling complex phenomena and solving various statistical problems.

Transformations are used for several key reasons:

• Simplification: Sometimes, a transformed variable might have a simpler distribution (e.g., a normal distribution) that is easier to work with.
• Modeling: Many real-world phenomena can be modeled by transforming existing random variables. For instance, converting a variable from Celsius to Fahrenheit is a linear transformation.
• Deriving New Distributions: We often need the distribution of a function of one or more random variables, such as the sum of two independent random variables or the ratio of two random variables.

There are several common methods to find the distribution of a transformed random variable, depending on whether the original variable is discrete or continuous, and the nature of the transformation function.

This is a general method applicable to both discrete and continuous random variables. If $Y = g(X)$, where $X$ is a random variable with CDF $F_X(x)$, we can find the CDF of $Y$, $F_Y(y)$, by using the relationship $F_Y(y) = P(Y \le y) = P(g(X) \le y)$. By solving the inequality $g(X) \le y$ for $X$, we can express this probability in terms of $F_X(x)$.

Once the CDF of $Y$ is found, its PDF (for continuous variables) or PMF (for discrete variables) can be obtained by differentiation or differencing, respectively. For a continuous random variable $X$ with PDF $f_X(x)$ and a strictly monotonic transformation $Y = g(X)$, the PDF of $Y$ is given by f_Y(y) = f_X(g^{-1}(y)) \left| \frac{d}{dy} g^{-1}(y) ight|, where $g^{-1}$ is the inverse function of $g$.

When dealing with functions of multiple random variables, such as $Z = h(X, Y)$, the Jacobian method is often employed for continuous variables. This method generalizes the single-variable PDF transformation formula and involves the determinant of the Jacobian matrix of the transformation. For discrete variables, we directly compute the probability $P(Z=z) = P(h(X,Y)=z)$ by summing probabilities over all pairs $(x,y)$ that satisfy $h(x,y)=z$.

Several standard transformations lead to well-known distributions:

• Linear Transformations: If $Y = aX + b$, where $a$ and $b$ are constants. If $X \sim N(\mu, \sigma^2)$, then $Y \sim N(a\mu + b, a^2\sigma^2)$.
• Sum of Independent Gamma Variables: If X_1 \sim Gamma(\alpha_1, eta) and X_2 \sim Gamma(\alpha_2, eta) are independent, then X_1 + X_2 \sim Gamma(\alpha_1 + \alpha_2, eta).
• Ratio of Independent Normal Variables: The ratio of two independent standard normal random variables follows a Cauchy distribution.

For actuarial exams, mastering transformations is essential. Focus on:

• Understanding the CDF and PDF methods thoroughly.
• Recognizing common transformations and their resulting distributions.
• Practicing problems involving both single and multiple random variable transformations.
• Being comfortable with the Jacobian method for continuous variables.