Suppose X and Y are independent random variables, each with...

Suppose $X$ and $Y$ are independent random variables, each with the density function $f$ given by $$ f(x)= \begin{cases}1 & \text { for } 0<x<1 \\ 0 & \text { otherwise. }\end{cases} $$ Show that the density function of $Z=X+Y$ is given by $$ f_Z(z)= \begin{cases}z & \text { for } 0 \leq z \leq 1 \\ 2-z & \text { for } 1<z \leq 2 \\ 0 & \text { otherwise. }\end{cases} $$ Thus, $Z$ has a triangular distribution. (As we shall see in Chapter 3, $X$ and $Y$ are said to be uniformly distributed.) Hint: This Exercise can be solved by using convolution (Theorem 2.7.5) or by using the Laplace-Stieltjes transform (Theorem 2.9.3(a) and (d)).

Suppose $X$ and $Y$ are independent random variables, each with the density function $f$ given by
$$
f(x)= \begin{cases}1 & \text { for } 0<x<1 \\ 0 & \text { otherwise. }\end{cases}
$$
Show that the density function of $Z=X+Y$ is given by
$$
f_Z(z)= \begin{cases}z & \text { for } 0 \leq z \leq 1 \\ 2-z & \text { for } 1<z \leq 2 \\ 0 & \text { otherwise. }\end{cases}
$$
Thus, $Z$ has a triangular distribution. (As we shall see in Chapter 3, $X$ and $Y$ are said to be uniformly distributed.) Hint: This Exercise can be solved by using convolution (Theorem 2.7.5) or by using the Laplace-Stieltjes transform (Theorem 2.9.3(a) and (d)).

Key Concepts

Uniform Distribution

A uniform distribution is a type of probability distribution in which all outcomes in a given interval are equally likely. For continuous random variables, this often means the probability density function is constant over the interval and zero elsewhere. In the context of the problem, both X and Y have the same uniform distribution, which is the foundation for identifying how their sum behaves.

Independence of Random Variables

Independence between random variables means that the joint probability density function can be expressed as the product of the individual densities. This property is crucial when calculating the distribution of the sum of two random variables because it allows one to use convolution, knowing that the variables do not affect each other.

Convolution of Probability Density Functions

Convolution is a mathematical operation used to derive the distribution of the sum of two independent random variables. It involves integrating the product of one random variable’s density function with the shifted density of the other. This method provides a systematic way to compute the resulting density, which in this problem results in a triangular distribution for the variable Z.

Triangular Distribution

A triangular distribution is characterized by a density function that increases linearly to a peak (mode) and then decreases linearly. It often arises as the sum of two independent uniform distributions. In this problem, the result of convolving two uniform (0,1) random variables produces a density function that is triangular, demonstrating the transition from a uniform to a triangular shape in the sum's distribution.

Laplace-Stieltjes Transform

The Laplace-Stieltjes transform is a tool used in probability theory to study distributions through an integral transform. It can simplify the process of finding the distribution of a sum of random variables by transforming convolution operations into products in the transform domain. This method provides an alternative approach to convolution for solving the problem at hand.

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