Suppose 𝑋 is a normal variable with mean 𝜇 and standard deviation 𝜎. It is given that 𝑃 (𝑋 > 125) = 0.25 and that 𝑃 (𝑋 > 95) = 0.87. Find 𝜇 and 𝜎. (8 marks)

To solve for the mean (\(\mu\)) and standard deviation (\(\sigma\)) of the normal variable \(X\), given that \(P(X > 125) = 0.25\) and \(P(X > 95) = 0.87\), we'll follow these steps:

### Step 1: Convert Probabilities to Standard Normal Form
The probability statements are given in terms of \(X\). We first need to convert these into the standard normal form \(Z\) using the transformation:
\[
Z = \frac{X - \mu}{\sigma}
\]
This allows us to use the standard normal distribution table (Z-table).

### Step 2: Express the Probabilities Using Z-Scores
1. **For \(P(X > 125) = 0.25\):**
   \[
   P(X > 125) = P\left(\frac{X - \mu}{\sigma} > \frac{125 - \mu}{\sigma}\right) = 0.25
   \]
   Using the Z-table, \(P(Z > z)\) corresponds to \(z = 0.6745\) when \(P(Z > z) = 0.25\).
   \[
   \frac{125 - \mu}{\sigma} = 0.6745
   \]
   \[
   125 - \mu = 0.6745\sigma \quad \text{(Equation 1)}
   \]

2. **For \(P(X > 95) = 0.87\):**
   \[
   P(X > 95) = P\left(\frac{X - \mu}{\sigma} > \frac{95 - \mu}{\sigma}\right) = 0.87
   \]
   Using the Z-table, \(P(Z > z)\) corresponds to \(z = -1.1262\) when \(P(Z > z) = 0.87\).
   \[
   \frac{95 - \mu}{\sigma} = -1.1262
   \]
   \[
   95 - \mu = -1.1262\sigma \quad \text{(Equation 2)}
   \]

### Step 3: Solve the System of Equations
We have the following system of equations:
\[
125 - \mu = 0.6745\sigma \quad \text{(Equation 1)}
\]
\[
95 - \mu = -1.1262\sigma \quad \text{(Equation 2)}
\]

Subtract Equation 2 from Equation 1:
\[
(125 - \mu) - (95 - \mu) = 0.6745\sigma - (-1.1262\sigma)
\]
\[
30 = 1.8007\sigma
\]
\[
\sigma = \frac{30}{1.8007} \approx 16.66
\]

Now, substitute \(\sigma \approx 16.66\) back into either Equation 1 or Equation 2 to find \(\mu\). Using Equation 1:
\[
125 - \mu = 0.6745 \times 16.66
\]
\[
125 - \mu \approx 11.24
\]
\[
\mu \approx 125 - 11.24 \approx 113.76
\]

### Final Answer:
- Mean (\(\mu\)) ≈ 113.76
- Standard Deviation (\(\sigma\)) ≈ 16.66

These are the required values of \(\mu\) and \(\sigma\).

Would you like further details or have any questions?

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Here are 5 related questions you might find interesting:
1. How would the solution change if the given probabilities were different?
2. What is the significance of the Z-score in normal distribution problems?
3. Can the value of \(\sigma\) ever be negative?
4. How can you verify the solution using a graph or a statistical software tool?
5. What is the impact of changing the standard deviation on the distribution curve?

**Tip:** Remember that Z-scores standardize a normal distribution, making it easier to use the standard normal table for calculations.

Learn how to calculate the mean (𝜇) and standard deviation (𝜎) of a normal variable using given probabilities. Solve for 𝜇 and 𝜎 with step-by-step instructions and formula applications. Suitable for university-level students and professionals.

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