Product of Numbers from a Log Normal Distribution

Let $X_i$,  $i \in \{1, 2, 3, 4\}$ be independent log-normally distributed variables with:

$$X_i \sim \textrm{LogNormal}\left(\frac{1}{2}, 1^2\right)$$

Now let $Z = X_1 X_2 X_3 X_4$, the product of the four iid random variables. We seek to determine $P(Z \geq 50)$, the probability that the product has value at least 50. 

Note a key fact about the log-normal distribution: that if $X$ is log-normally distributed, then $\log X$ is normally distributed. We can write this as:

$$\log X_i \sim \textrm{Normal}\left(\frac{1}{2}, 1^2\right)$$

We use this fact to determine the distribution of $Z$. We look at $\log Z$:

$$\log Z = \log\left( X_1 X_2 X_3 X_4 \right) = \log X_1 + \log X_2 + \log X_3 + \log X_4$$

Based on our fact above, each of $\log X_i$ is normally distributed. Since the sum of normally distributed variabales is also normally distributed with their means and variances summed, we get:

$$\log Z \sim \textrm{Normal}\left( 2, 2^2\right)$$

Since $\log Z$ is normally distributed, then $Z$ is log-normally distributed, we get:

$$Z \sim \textrm{LogNormal}\left( 2, 2^2\right)$$

We use the CDF of the log-normal distribution to compute directly:

$$\boxed{P(Z \geq 50) = 0.1695}$$