Step-by-Step Calculation of the Expected Value

We are given:

• $X \sim \mathcal{N}(\mu, \sigma^2)$
• We need to find $\mathbb{E}(-X \mid X < \mu + 3\sigma \cdot \Phi^{-1}(p))$, where $\Phi^{-1}(p)$ is the inverse CDF (quantile) of the standard normal distribution at some probability $p$.

1. Standardization of $X$

First, let's standardize the random variable $X$:

$Z = \frac{X - \mu}{\sigma}$

where $Z \sim \mathcal{N}(0, 1)$ (i.e., $Z$ is a standard normal random variable).

2. Threshold in Terms of $Z$

The threshold condition $X < \mu + 3\sigma \cdot \Phi^{-1}(p)$ can be rewritten in terms of $Z$:

$X < \mu + 3\sigma \cdot \Phi^{-1}(p)$ $\Rightarrow Z < \frac{\mu + 3\sigma \cdot \Phi^{-1}(p) - \mu}{\sigma}$ $\Rightarrow Z < 3 \Phi^{-1}(p)$

Thus, the condition becomes $Z < 3 \Phi^{-1}(p)$.

3. Expected Value in Terms of $Z$

We want to find the expected value:

$\mathbb{E}(-X \mid Z < 3\Phi^{-1}(p))$

Using the standardization $X = \mu + \sigma Z$, we have:

$\mathbb{E}(-X \mid Z < 3\Phi^{-1}(p)) = -\mathbb{E}(\mu + \sigma Z \mid Z < 3\Phi^{-1}(p))$

Expanding the expectation:

$\mathbb{E}(-X \mid Z < 3\Phi^{-1}(p)) = -\left[\mu + \sigma \mathbb{E}(Z \mid Z < 3\Phi^{-1}(p))\right]$

4. Calculate $\mathbb{E}(Z \mid Z < 3\Phi^{-1}(p))$

The expectation $\mathbb{E}(Z \mid Z < 3\Phi^{-1}(p))$ for a standard normal variable $Z$ is known and given by:

$\mathbb{E}(Z \mid Z < c) = -\frac{\phi(c)}{\Phi(c)}$

where $\phi(c)$ is the probability density function (PDF) of the standard normal distribution evaluated at $c$, and $\Phi(c)$ is the CDF evaluated at $c$. In our case, $c = 3\Phi^{-1}(p)$.

So:

$\mathbb{E}(Z \mid Z < 3\Phi^{-1}(p)) = -\frac{\phi(3\Phi^{-1}(p))}{\Phi(3\Phi^{-1}(p))}$

5. Final Expression for $\mathbb{E}(-X \mid Z < 3\Phi^{-1}(p))$

Substitute the result back into the equation for the expected value:

$\mathbb{E}(-X \mid Z < 3\Phi^{-1}(p)) = -\left[\mu - \sigma \cdot \frac{\phi(3\Phi^{-1}(p))}{\Phi(3\Phi^{-1}(p))}\right]$

Simplifying:

$\mathbb{E}(-X \mid Z < 3\Phi^{-1}(p)) = -\mu + \sigma \cdot \frac{\phi(3\Phi^{-1}(p))}{\Phi(3\Phi^{-1}(p))}$

Summary

The expected value is given by:

$\mathbb{E}(-X \mid X < \mu + 3\sigma \cdot \Phi^{-1}(p)) = -\mu + \sigma \cdot \frac{\phi(3\Phi^{-1}(p))}{\Phi(3\Phi^{-1}(p))}$

Questions:

1. Would you like to calculate this explicitly for a specific value of $p$?
2. Do you need help with understanding the standard normal PDF $\phi(z)$ and CDF $\Phi(z)$?
3. Are you interested in the implications of this result for different values of $p$?
4. Would you like to explore how this expectation changes with different $\mu$ and $\sigma$?
5. Do you want to see a graphical representation of this conditional expectation?

Tip:

Understanding how to manipulate expectations under conditional probabilities is crucial in fields such as statistics and financial mathematics, especially when dealing with risk management and insurance models.