What Is the MGF of the Ratio X/Y?

There is no universal MGF for the ratio $R=X/Y$. Its distribution depends on the joint behavior of $X$ and $Y$, and the expectation defining its MGF may be infinite for every nonzero value of the argument.

The general expression is $M_R(t)=\mathbb{E}[e^{tX/Y}]$, provided $Y\ne 0$ with probability 1 and the expectation is finite. Here is how to find the ratio distribution when densities are available, and why the MGF may still fail to exist.

The general formula

A random variable’s moment-generating function is $M_R(t)=\mathbb{E}[e^{tR}]$. For $R=X/Y$, this becomes

This expression is useful only when its expectation is finite. By the usual definition, an MGF must be finite for all $t$ in some open interval containing zero. A value at $t=0$ alone, where $M_R(0)=1$, does not establish that an MGF exists.

If $(X,Y)$ has joint density $f_{X,Y}(x,y)$ and $P(Y=0)=0$, the ratio has density

The factor $|y|$ is the absolute Jacobian of the transformation $x=ry$. Once you have this density, test whether

exists and is finite around $t=0$. If $X$ and $Y$ are independent with densities, you can use $f_{X,Y}(x,y)=f_X(x)f_Y(y)$ in the ratio-density formula. Independence is not required, but the joint distribution is needed.

Worked example: a ratio of standard normals

Let $X$ and $Y$ be independent standard normal random variables. Each has an MGF, $e^{t^2/2}$, but their ratio does not.

Using the density transformation,

where $\phi$ is the standard normal density. This is the standard Cauchy density. For $t>0$, the integral $\int e^{tr}/[\pi(1+r^2)]\,dr$ diverges in the positive tail. For $t<0$, it diverges in the negative tail. Thus $M_R(t)$ is infinite for every $t\ne 0$, even though both variables in the ratio have MGFs.

A positive ratio can also lack an MGF

Let $X$ and $Y$ be independent exponential random variables with rate 1. Both are positive, and the density of $R=X/Y$ is

For $t>0$, the exponential factor $e^{tr}$ makes the MGF integral diverge. For $t<0$, the integral is finite, but that gives only a one-sided range of finite values, not an open interval around zero. So this ratio does not have an MGF under the usual definition either.

Common mistakes

• Using the separate MGFs of $X$ and $Y$. The MGF of a ratio is not obtained by dividing one MGF by another. The ratio depends on the joint distribution.
• Assuming independence without being told. Marginal distributions alone do not determine the distribution of $X/Y$; dependence can change it.
• Treating $M_R(0)=1$ as proof of existence. The expectation must be finite in a neighborhood of zero, not just at zero.
• Ignoring a possible zero denominator. If $P(Y=0)>0$, the ratio is not defined on an event with positive probability unless a special convention is specified.
• Confusing an MGF with a characteristic function. The characteristic function $\mathbb{E}[e^{itR}]$ exists for every real $t$ because its integrand has absolute value 1. That does not imply the MGF exists.

To answer a particular ratio problem, first specify the joint law (or independence and both marginal laws), then derive the ratio distribution or evaluate $\mathbb{E}[e^{tX/Y}]$, and finally check finiteness on both sides of zero.