Sum of independent random variables

Let $ X$ and $ Y$ be independent random variables having the respective probability density functions $ f_X(x)$ and $ f_Y(y)$. Then the cumulative distribution function $ F_Z(z)$ of the random variable $ Z = X + Y$ can be given as follows.

$\displaystyle F_Z(z) = P(X + Y \le z)
= \int_{-\infty}^\infty\left[\int_{-\inft...
...-x}
f_X(x) f_Y(y) \,dy\right]\,dx
= \int_{-\infty}^\infty f_X(x) F_Y(z-x)\,dx,
$

where $ F_Y$ is the cdf of $ Y$. By differentiating $ F_Z(z)$, we can obtain the pdf $ f_Z(z)$ of $ Z$ as

$\displaystyle f_Z(z) = \frac{d}{dz} F_Z(z)
= \int_{-\infty}^\infty f_X(x)
\left(\frac{d}{dz} F_Y(z-x) \right) \,dx
= \int_{-\infty}^\infty f_X(x) f_Y(z-x) \,dx.
$

The form of integration $ \int_{-\infty}^\infty f(x) g(z-x) \,dx$ is called the convolution. Thus, the pdf $ f_Z(z)$ is given by the convolution of the pdf's $ f_X(x)$ and $ f_Y(y)$. However, the use of moment generating function makes it easier to ``find the distribution of the sum of independent random variables.''

Let $ X$ and $ Y$ be independent normal random variables with the respective parameters $ (\mu_x,\sigma_x^2)$ and $ (\mu_y,\sigma_y^2)$. Then the sum $ Z = X + Y$ of random variables has the mgf

$\displaystyle M_Z(t) = M_X(t) \cdot M_Y(t)
= \exp\left( \frac{\sigma_x^2 t^2}{2...
...\exp\left( \frac{(\sigma_x^2 + \sigma_y^2) t^2}{2}
+ (\mu_x + \mu_y) t \right)
$

which is the mgf of normal distribution with parameter $ (\mu_x+\mu_y, \sigma_x^2 + \sigma_y^2)$. By the property (a) of mgf, we can find that $ Z$ is a normal random variable with parameter $ (\mu_x+\mu_y, \sigma_x^2 + \sigma_y^2)$.

Let $ X$ and $ Y$ be independent gamma random variables with the respective parameters $ (\alpha_1,\lambda)$ and $ (\alpha_2,\lambda)$. Then the sum $ Z = X + Y$ of random variables has the mgf

$\displaystyle M_Z(t) = M_X(t) \cdot M_Y(t)
= \left(\frac{\lambda}{\lambda-t}\ri...
...ght)^{\alpha_2}
= \left(\frac{\lambda}{\lambda-t}\right)^{\alpha_1 + \alpha_2}
$

which is the mgf of gamma distribution with parameter $ (\alpha_1+\alpha_2,\lambda)$. Thus, $ Z$ is a gamma random variable with parameter $ (\alpha_1+\alpha_2,\lambda)$.



Generated by MATH GO: 2006-02-27