Example 9.7.3. Exponential X vs Normal \(\overline{X}\).
Consider an exponential variable X with mean time till first success of \(\mu = 4\text{.}\) Then, \(\sigma = 2\) using the exponential formulas.
You can use the exponential probability function to compute probabilities dealing with X. Indeed,
\begin{equation*}
P(X \lt 3.9) = F(3.9) = 1 - e^{-3.9/4} \approx 0.6228 .
\end{equation*}
If instead you plan to sample from this distribution n=32 times, the Central Limit Theorem implies that you will get a random variable \(\overline{X}\) which has an approximate normal distribution with the same mean but with new variance \(\sigma_{\overline{X}}^2 = \frac{4}{32} = \frac{1}{8}\text{.}\) Therefore
\begin{equation*}
P( \overline{X} \lt 3.9 ) \\ \approx \text{normalcdf}(0,3.9,4,sqrt(1/8)) = 0.3886 .
\end{equation*}