Suppose that X is a ramdom variable for which `E(X)=10` and Var (X) = 25. Find the values of a and b such that Y = ax - b has expectation 0 and variance 1.

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that the random variable \( Y = aX - b \) has an expectation of 0 and a variance of 1, given that \( E(X) = 10 \) and \( \text{Var}(X) = 25 \).

### Step-by-Step Solution:

**Step 1: Understanding the Expectation of Y**

The expectation of \( Y \) can be expressed in terms of \( X \):
\[ E(Y) = E(aX - b) = aE(X) - b \]
Given \( E(X) = 10 \), we substitute this into the equation:
\[ E(Y) = a \cdot 10 - b \]
We want \( E(Y) = 0 \), so we set up the equation:
\[ a \cdot 10 - b = 0 \]
This simplifies to:
\[ b = 10a \tag{1} \]

**Step 2: Understanding the Variance of Y**

The variance of \( Y \) is given by:
\[ \text{Var}(Y) = \text{Var}(aX - b) = a^2 \text{Var}(X) \]
Since \( \text{Var}(X) = 25 \), we have:
\[ \text{Var}(Y) = a^2 \cdot 25 \]
We want \( \text{Var}(Y) = 1 \), so we set up the equation:
\[ a^2 \cdot 25 = 1 \]
Dividing both sides by 25 gives:
\[ a^2 = \frac{1}{25} \]
Taking the square root of both sides, we find:
\[ a = \frac{1}{5} \quad \text{or} \quad a = -\frac{1}{5} \]

**Step 3: Finding the Corresponding b Value**

Using equation (1), we substitute \( a = \frac{1}{5} \) into it to find \( b \):
\[ b = 10 \cdot \frac{1}{5} = 2 \]
If we take \( a = -\frac{1}{5} \):
\[ b = 10 \cdot -\frac{1}{5} = -2 \]

Thus, we have two possible pairs:
1. \( (a, b) = \left(\frac{1}{5}, 2\right) \)
2. \( (a, b) = \left(-\frac{1}{5}, -2\right) \)

### Final Answer:
The values of \( a \) and \( b \) such that \( Y = aX - b \) has expectation 0 and variance 1 are:
- \( a = \frac{1}{5}, b = 2 \) or \( a = -\frac{1}{5}, b = -2 \).