Question

Suppose that \({{\bf{X}}_{\bf{1}}}\;{\bf{and}}\;{{\bf{X}}_{\bf{2}}}\) are i.i.d. random variables andthat each of them has a uniform distribution on theinterval [0, 1]. Find the p.d.f. of\({\bf{Y = }}{{\bf{X}}_{\bf{1}}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}\).

Short answer

The p.d.f of \(Y = {X_1} + {X_2}\) is,

\(g\left( y \right) = \left\{ \begin{array}{l}y\;\;\;\;\;\;\;\;\;\;\;for\;0 < y \le 1\\2 - y\;\;\;\;\;\;for\;1 < y < 2\\0\;\;\;\;\;\;\;\;\;\;\;{\rm{otherwise}}\end{array} \right.\)

Step-by-step solution

Given information

\({X_1},{X_2}\) is a random variable following uniform distribution on a given interval [0,1], that is,\({X_i} \sim U[0,1]\;\;\;\forall \;i = 1,2\).

State p.d.f of

The pdf of any random variable X with uniform distribution is obtained by using the formula: \(\frac{1}{{b - a}};a \le x \le b\).

Here, \(a = 0,b = 1\).

Therefore, the pdf of \({X_i} \sim U[0,1]\) is expressed as,

\({f_{{x_i}}} = \left\{ \begin{array}{l}\frac{1}{{1 - 0}} = 1\;\;\;\;\;\;\;\;\;\;0 \le {x_i} \le 1\\0;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{array} \right.\)

Define new variables Y and Z 

Define,

\(\begin{aligned}{\bf{Y = }}{{\bf{X}}_{\bf{1}}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}\\{\bf{Z = }}{{\bf{X}}_{\bf{1}}}{\bf{ - }}{{\bf{X}}_{\bf{2}}}\end{aligned}\)

Obtain the two uniformly distributed variables in terms of the new variables and obtain the ranges as follows.

From definitions of Y and Z,

If \({x_1} = 0 \Rightarrow y + z = 0\)

If \({x_1} = \frac{{y + z}}{2} \Rightarrow z = - y\)

Then,

\({x_2} = 0 \Rightarrow y = z\)

\({x_2} = \frac{{y - z}}{2}\)which implies that,

\(\begin{aligned}{x_1} &= 1 \Rightarrow y + z = 2\\{x_2} &= 1 \Rightarrow y - z = 2\end{aligned}\)

Perform the Jacobian Transformation

The jacobian is obtained as follows,

\(\begin{aligned}J &= \left| {\begin{aligned}{}{\frac{{\partial {x_1}}}{{\partial Y}}}&{\frac{{\partial {x_2}}}{{\partial Y}}}\\{\frac{{\partial {x_1}}}{{\partial Z}}}&{\frac{{\partial {x_2}}}{{\partial Z}}}\end{aligned}} \right|\\ &= \left| {\begin{aligned}{}{\frac{1}{2}}&{\frac{1}{2}}\\{\frac{1}{2}}&{\frac{{ - 1}}{2}}\end{aligned}} \right|\\ &= \frac{1}{2}\end{aligned}\)

Define the joint pdf of the two variables Y and Z

Define the joint distribution as,

\(\begin{aligned}g\left( {y,z} \right) &= f\left( {{x_1},{x_2}} \right)\left| J \right|\\ &= \frac{1}{2},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0 < y < 2, - 1 < z < 1\end{aligned}\)

From the joint distribution, the marginal pdf of the variables is obtained as,

\(\begin{aligned}{g_1}\left( y \right) &= \int_{ - y}^y {\frac{1}{2}dz} \;\\ &= \frac{1}{2}\left| z \right|_{ - y}^y\\ &= y\end{aligned}\)

\(\begin{aligned}{g_2}\left( z \right) &= \int_{y - 2}^{2 - y} {\frac{1}{2}dz} \;\\ &= \frac{1}{2}\left| z \right|_{y - 2}^{2 - y}\\ &= 2 - y\end{aligned}\)

Thus, the distribution of \(Y = {X_1} + {X_2}\)is,

\(g\left( y \right) = \left\{ \begin{array}{l}y\;\;\;\;\;\;\;\;\;\;\;for\;0 < y \le 1\\2 - y\;\;\;\;\;\;for\;1 < y < 2\\0\;\;\;\;\;\;\;\;\;\;\;{\rm{otherwise}}\end{array} \right.\)