Grinstead-Snell-Probability · Problem 1

Law of Large Numbers

Mathematics · convolution-density-functions

Problem

Let

X

and

Y

be independent real-valued random variables with density functions

f_X(x)

and

f_Y(y)

, respectively. Show that the density function of the sum

X + Y

is the convolution of the functions

f_X(x)

and

f_Y(y)

. Hint: Let

\bar{X}

be the joint random variable

(X, Y)

. Then the joint density function of

\bar{X}

f_X(x)f_Y(y)

, since

X

and

Y

are independent. Now compute the probability that

X + Y \leq z

, by integrating the joint density function over the appropriate region in the plane. This gives the cumulative distribution function of

Z

. Now differentiate this function with respect to

z

to obtain the density function of

z