Understanding the Normal Distribution and Conditional Probability

The question asks for a conditional probability involving a normally distributed variable \(X\). We are given that \(X \sim N(\mu, \sigma^2)\), meaning \(X\) follows a normal distribution with mean \(\mu\) and variance \(\sigma^2\). A key condition given is that \(\mu^2 = \sigma^2\) and \(\mu > 0\). Since \(\mu > 0\) and \(\sigma\) represents the standard deviation (which must be non-negative), \(\sigma = \sqrt{\mu^2} = |\mu|\). Given \(\mu > 0\), we have \(\sigma = \mu\).

We need to find the value of the conditional probability \(P(X < -\mu | X < \mu)\). The formula for conditional probability is \(P(A|B) = \frac{P(A \cap B)}{P(B)}\).

In this case, event \(A\) is \(X < -\mu\) and event \(B\) is \(X < \mu\).

Let's analyze the intersection of these two events, \(A \cap B\):

• Event A: \(X\) is less than \(-\mu\).
• Event B: \(X\) is less than \(\mu\).

Since \(\mu > 0\), we know that \(-\mu < \mu\). Therefore, if \(X < -\mu\), it is automatically true that \(X < \mu\). So, the event \(X < -\mu\) is a subset of the event \(X < \mu\). This means \(A \cap B\) is simply the event \(X < -\mu\).

Thus, the conditional probability simplifies to:

\[P(X < -\mu | X < \mu) = \frac{P(X < -\mu)}{P(X < \mu)}\]

Standardizing the Normal Distribution

To find these probabilities, we need to standardize the variable \(X\). We transform \(X\) into a standard normal variable \(Z\) using the formula \(Z = \frac{X - \mu}{\sigma}\). Since we found that \(\sigma = \mu\), the standardization formula becomes \(Z = \frac{X - \mu}{\mu}\).

Let's find \(P(X < -\mu)\):

• Start with the inequality: \(X < -\mu\)
• Subtract \(\mu\) from both sides: \(X - \mu < -\mu - \mu\)
• Simplify: \(X - \mu < -2\mu\)
• Divide by \(\mu\) (which is positive, so the inequality direction doesn't change): \(\frac{X - \mu}{\mu} < \frac{-2\mu}{\mu}\)
• Substitute with \(Z\): \(Z < -2\)

So, \(P(X < -\mu) = P(Z < -2)\).

Now, let's find \(P(X < \mu)\):

• Start with the inequality: \(X < \mu\)
• Subtract \(\mu\) from both sides: \(X - \mu < \mu - \mu\)
• Simplify: \(X - \mu < 0\)
• Divide by \(\mu\): \(\frac{X - \mu}{\mu} < \frac{0}{\mu}\)
• Substitute with \(Z\): \(Z < 0\)

So, \(P(X < \mu) = P(Z < 0)\).

Using the Standard Normal Cumulative Distribution Function

The cumulative distribution function (CDF) for the standard normal distribution, denoted by \(P(Z \le z)\), gives the probability \(P(Z < z)\) (since the normal distribution is continuous, \(P(Z < z) = P(Z \le z)\)).

For the standard normal distribution, which is symmetric about 0, the probability of \(Z\) being less than 0 is 0.5. So, \(P(Z < 0) = 0.5\).

For the probability \(P(Z < -2)\), we can use the symmetry property: \(P(Z < -z) = P(Z > z)\). Also, the total probability under the curve is 1, so \(P(Z > z) = 1 - P(Z \le z)\). Therefore, \(P(Z < -2) = P(Z > 2) = 1 - P(Z \le 2)\).

Calculating the Conditional Probability

Now we substitute these probabilities back into the conditional probability formula:

\[P(X < -\mu | X < \mu) = \frac{P(Z < -2)}{P(Z < 0)}\]

\[= \frac{1 - P(Z \le 2)}{0.5}\]

\[= 2[1 - P(Z \le 2)]\]

This result matches option 4.

Revision Table: Key Concepts

Additional Information on Normal Distribution and Conditional Probability

The normal distribution is one of the most important probability distributions in statistics. Its shape is determined by its mean (\(\mu\)) and standard deviation (\(\sigma\)). The standard normal distribution is a special case that is widely used because any normal distribution can be transformed into it through standardization.

Conditional probability helps us update our belief about the likelihood of an event happening based on the knowledge that another event has already occurred. In this problem, knowing that \(X < \mu\) provides context for evaluating the probability that \(X < -\mu\).

The condition \(\mu^2 = \sigma^2\) with \(\mu > 0\) is crucial as it tells us that the standard deviation is equal to the mean. This specific relationship allows us to simplify the standardization step and arrive at the numerical values needed for Z.

The cumulative distribution function \(P(Z \le z)\), often denoted by \(\Phi(z)\), is a fundamental tool for working with the standard normal distribution. Tables or software are typically used to find values of \(\Phi(z)\) for different \(z\). The expression \(1 - P(Z \le 2)\) represents the area under the standard normal curve to the right of \(Z=2\).