Detailed calculation of the expected value

### 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.

Explore the detailed step-by-step calculation of the expected value under conditional probability using the normal distribution and standard normal distribution concepts. Learn how to apply the standardization of variables, calculate expectations under conditional probabilities, and interpret the results. Suitable for advanced mathematics students and professionals.

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