Monthly Archives: May, 2011

Examples of convolution (continuous case)

The method of convolution is a great technique for finding the probability density function (pdf) of the sum of two independent random variables. We state the convolution formula in the continuous case as well as discussing the thought process. Some examples are provided to demonstrate the technique and are followed by an exercise.

The Convolution Formula (Continuous Case)
Let X and Y be independent continuous random variables with pdfs f_X(x) and f_Y(y), respectively. Let Z=X+Y. Then the following is the pdf of Z.

\displaystyle \begin{aligned}f_Z(z)&=\int_{-\infty}^\infty f_Y(z-x) f_X(x) \ dx \\&=\int_{-\infty}^\infty f_X(z-y) f_Y(y) \ dy  \end{aligned}

Let’s look at the thought process behind the formula. Since X and Y are independent, the joint pdf of X and Y is f_{X,Y}(x,y)=f_X(x) \ f_Y(y). The pdf of Z is simply the sum of the “joint density” at the points of the line z=x+y. In Figure 1 below, every point at the line z=x+y is of the form (x,z-x). The joint density at each such point is f_X(x) \ f_Y(z-x). Summing the values of these joint density produces the probability density function of Z.

Setting the limits of the integral depends on knowing the range of possible values of x or y for a given line z=x+y. If X and Y can only take on positive values, then for a given line z=x+y, both x or y can range from 0 to z (see Example 1 below).

Example 1
Let X and Y be independent exponentially distributed variables with common density f(x)=\alpha e^{-\alpha x} where \alpha>0. Then the following is the pdf of Z=X+Y.

\displaystyle \begin{aligned}f_Z(z)&=\int_{0}^z f_Y(z-x) f_X(x) \ dx \\&=\int_{0}^z \alpha e^{-\alpha (z-x)} \alpha e^{-\alpha x} \ dx \\&=\int_{0}^z \alpha^2 e^{-\alpha z} \ dx =\alpha^2 \ z \ e^{-\alpha z}  \end{aligned}

The above pdf indicates that the independent sum of two identically distributed exponential variables has a Gamma distribution with parameters 2 and \alpha.

Example 2
Let X and Y be independent uniformly distributed variables, U(0,10) and U(0,20), respectively. The pdf of Z=X+Y is:

\displaystyle f_Z(z)=\left\{\begin{matrix} \displaystyle \frac{z}{200}&\ \ \ \ \ \ 0 \le z <10 \\{\text{ }}& \\{\displaystyle \frac{1}{20}}&\ \ \ \ \ \ 10 \le z <20 \\{\text{ }}& \\{\displaystyle \frac{30-z}{200}}&\ \ \ \ \ \ 20 \le z <30  \end{matrix}\right.

The convolution formula is applied three times. For the first case, the line x+y=z ranges in 0 \le z < 10. For each such line, we have 0<x<z. Figure 2 below is a representative diagram.

\displaystyle \begin{aligned}f_Z(z)&=\int_{0}^{z} f_Y(z-x) f_X(x) \ dx \\&=\int_{0}^{z} \frac{1}{20} \frac{1}{10} \ dx =\frac{z}{200}  \end{aligned}

For the second case, the line x+y=z ranges in 10 \le z < 20. For each such line, x ranges from 0 to 10. Figure 3 below is a representative diagram.

\displaystyle \begin{aligned}f_Z(z)&=\int_{0}^{10} f_Y(z-x) f_X(x) \ dx \\&=\int_{0}^{10} \frac{1}{10} \frac{1}{20} \ dx =\frac{1}{20}  \end{aligned}

For the third case, the line x+y=z ranges in 20 \le z < 30. Figure 4 below is a representative diagram.

\displaystyle \begin{aligned}f_Z(z)&=\int_{z-20}^{10} f_Y(z-x) f_X(x) \ dx \\&=\int_{z-20}^{10} \frac{1}{10} \frac{1}{20} \ dx =\frac{30-z}{200}  \end{aligned}

The following is the graph of the pdf of Z=X+Y.

Exercise
Suppose that X is an exponentially distributed variable with pdf f(x)=\alpha e^{-\alpha x} and Y has the uniform distribution U(0,1). Find the pdf of the independent sum Z=X+Y.

\displaystyle f_Z(z)=\left\{\begin{matrix} \displaystyle 1-e^{-\alpha z}&\ \ \ \ \ \ 0 \le z <1 \\{\text{ }}& \\{\text{ }}& \\{\displaystyle e^{-\alpha z}(e^\alpha-1)}&\ \ \ \ \ \ 1 \le z <\infty  \end{matrix}\right.

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Independent sum of uniform

Consider the folowing problems.

  • Problem A. Suppose that the losses under two insurance policies are independent such that the loss from each policy follows a continuous uniform distribution on the interval from 0 to 2. Find the probability density function and the cumulative distribution function of the sum of the losses.
  • Problem B. Suppose that the losses under two insurance policies are independent such that the loss from one policy follows a continuous uniform distribution on the interval from 0 to 10 and the loss from the other policy is continuously uniformly distributed on the interval from 0 to 20. Find the probability density function and the cumulative distribution function of the sum of the losses.

Discussion of Problem A
Finding information about the probability distribution of the sum of several independent random variables is a basic topic in probability theory and is an important one in the SOA Exam P. One elementary method in finding the probability distribution of an independent sum is through the moment generating functions (e.g. independent sum of normal distributions). Another technique is convolution. For the independent sum of two uniform distributions, we can actually obtain the distribution by a graphical approach. The method of convolution can be applied to the examples discussed here.

Let X and Y be independent such that each follows U(0,2). Let Z=X+Y. Then the joint pdf of X and Y is \displaystyle f(x,y)=\frac{1}{4} over 0 \le x \le 2 and 0 \le y \le 2. The following is the cumulative distribution function of Z=X+Y:

\displaystyle F_Z(z)=\left\{\begin{matrix}0& \ \ \ \ \ \ z<0 \\{\text{ }}& \\{\displaystyle \frac{z^2}{8}}&\ \ \ \ \ \ 0 \le z <2 \\{\text{ }}& \\{\displaystyle 1-\frac{(4-z)^2}{8}}&\ \ \ \ \ \ 2 \le z <4 \\{\text{ }}& \\{1}&\ \ \ \ \ \ 4 \le z \end{matrix}\right.

For the case 0 \le z <2, F_Z(z) is obtained by multiplying the shaded area in the following diagram (Figure 1) by the joint pdf f(x,y)=\frac{1}{4}. Note that the area of the shaded region in Figure 1 is \displaystyle \frac{z^2}{2}.

For the case 2 \le z <4, F_Z(z) is obtained by multiplying the complement of the shaded area in the following diagram (Figure 2) by the joint pdf f(x,y)=\frac{1}{4}. Note that the area of the shaded region in Figure 2 is \displaystyle \frac{(4-z)^2}{2}. The complement of the shaded region is \displaystyle 4-\frac{(4-z)^2}{2}.

The following is the pdf of Z=X+Y.

\displaystyle f_Z(z)=\left\{\begin{matrix}\displaystyle \frac{z}{4}&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \le z <2 \\{\text{ }}& \\{\displaystyle \frac{4-z}{4}}&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2 \le z <4 \end{matrix}\right.

Problem B Answers
Suppose X and Y are U(0,10) and U(0,20), respectively. Suppose that Z=X+Y is an independent sum. Then the cdf and pdf of Z are:

\displaystyle F_Z(z)=\left\{\begin{matrix}0& \ \ \ \ \ \ z<0 \\{\text{ }}& \\{\displaystyle \frac{z^2}{400}}&\ \ \ \ \ \ 0 \le z <10 \\{\text{ }}& \\{\displaystyle \frac{z}{20}-\frac{1}{4}}&\ \ \ \ \ \ 10 \le z <20 \\{\text{ }}& \\{\displaystyle 1-\frac{(30-z)^2}{400}}&\ \ \ \ \ \ 20 \le z <30 \\{\text{ }}& \\{1}&\ \ \ \ \ \ 30 \le z \end{matrix}\right.

\displaystyle f_Z(z)=\left\{\begin{matrix} \displaystyle \frac{z}{200}&\ \ \ \ \ \ 0 \le z <10 \\{\text{ }}& \\{\displaystyle \frac{1}{20}}&\ \ \ \ \ \ 10 \le z <20 \\{\text{ }}& \\{\displaystyle \frac{30-z}{200}}&\ \ \ \ \ \ 20 \le z <30 \end{matrix}\right.