Let `X_1,X_2,….,X_(18)` be eighteen observations such that `sum_(i=1)^(18)(X_i-alpha)=36 and sum_(i=1)^(18)(X_i-beta)^2=90` , where `alpha and beta` are distinct real number. If the standard deviation of these observations is 1 then the value of `|alpha-beta|` is `"_____"`

Answer

To solve the problem, we will follow these steps:

### Step 1: Set up the equations
We are given two equations:
1. \(\sum_{i=1}^{18} (X_i - \alpha) = 36\)
2. \(\sum_{i=1}^{18} (X_i - \beta)^2 = 90\)

From the first equation, we can express the sum of the observations:
\[ \sum_{i=1}^{18} X_i - 18\alpha = 36 \implies \sum_{i=1}^{18} X_i = 36 + 18\alpha \]

### Step 2: Expand the second equation
Using the second equation:
\[ \sum_{i=1}^{18} (X_i - \beta)^2 = \sum_{i=1}^{18} (X_i^2 - 2X_i\beta + \beta^2) = 90 \]
This expands to:
\[ \sum_{i=1}^{18} X_i^2 - 2\beta \sum_{i=1}^{18} X_i + 18\beta^2 = 90 \]

### Step 3: Substitute \(\sum_{i=1}^{18} X_i\)
Substituting \(\sum_{i=1}^{18} X_i = 36 + 18\alpha\) into the second equation:
\[ \sum_{i=1}^{18} X_i^2 - 2\beta(36 + 18\alpha) + 18\beta^2 = 90 \]
Rearranging gives:
\[ \sum_{i=1}^{18} X_i^2 - 72\beta - 36\alpha\beta + 18\beta^2 = 90 \]

### Step 4: Use the standard deviation
The standard deviation of the observations is given as 1. The formula for variance is:
\[ \sigma^2 = \frac{\sum_{i=1}^{18} X_i^2}{n} - \left(\frac{\sum_{i=1}^{18} X_i}{n}\right)^2 \]
Substituting \(n = 18\) and \(\sigma^2 = 1\):
\[ 1 = \frac{\sum_{i=1}^{18} X_i^2}{18} - \left(\frac{36 + 18\alpha}{18}\right)^2 \]
This simplifies to:
\[ 1 = \frac{\sum_{i=1}^{18} X_i^2}{18} - \left(2 + \alpha\right)^2 \]

### Step 5: Rearranging the variance equation
Multiplying through by 18:
\[ 18 = \sum_{i=1}^{18} X_i^2 - 18(2 + \alpha)^2 \]
This gives:
\[ \sum_{i=1}^{18} X_i^2 = 18 + 18(2 + \alpha)^2 \]

### Step 6: Substitute back into the second equation
Now substitute \(\sum_{i=1}^{18} X_i^2\) back into the equation from Step 3:
\[ 18 + 18(2 + \alpha)^2 - 72\beta - 36\alpha\beta + 18\beta^2 = 90 \]
This simplifies to:
\[ 18(2 + \alpha)^2 - 72\beta - 36\alpha\beta + 18\beta^2 = 72 \]
Dividing through by 18:
\[ (2 + \alpha)^2 - 4\beta - 2\alpha\beta + \beta^2 = 4 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ (2 + \alpha)^2 - 4 = 4\beta + 2\alpha\beta - \beta^2 \]
This leads to:
\[ \alpha^2 + 4\alpha = 4\beta + 2\alpha\beta - \beta^2 \]

### Step 8: Solve for \(|\alpha - \beta|\)
We know that \(\alpha\) and \(\beta\) are distinct. The final equation can be manipulated to find \(|\alpha - \beta|\). After solving, we find:
\[ |\alpha - \beta| = 4 \]

### Final Answer
Thus, the value of \(|\alpha - \beta|\) is:
\[ \boxed{4} \]