A ramdom variable X can assume any positive integral value of n with a probability proportional to n with a probability proportional to `(1)/(3^(n))`. Find the expectation of X?

Answer

To find the expectation of the random variable \( X \) which can assume any positive integral value \( n \) with a probability proportional to \( \frac{1}{3^n} \), we will follow these steps:

### Step 1: Define the Probability Function
The probability \( P(X = n) \) is given to be proportional to \( \frac{1}{3^n} \). We can express this as:
\[ P(X = n) = c \cdot \frac{1}{3^n} \]
where \( c \) is a normalization constant.

### Step 2: Normalize the Probability Function
To find the constant \( c \), we need to ensure that the total probability sums to 1:
\[ \sum_{n=1}^{\infty} P(X = n) = 1 \]
Substituting the expression for \( P(X = n) \):
\[ \sum_{n=1}^{\infty} c \cdot \frac{1}{3^n} = 1 \]
This can be simplified to:
\[ c \cdot \sum_{n=1}^{\infty} \frac{1}{3^n} = 1 \]
The series \( \sum_{n=1}^{\infty} \frac{1}{3^n} \) is a geometric series with first term \( \frac{1}{3} \) and common ratio \( \frac{1}{3} \):
\[ \sum_{n=1}^{\infty} \frac{1}{3^n} = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{1/3}{2/3} = \frac{1}{2} \]
Thus, we have:
\[ c \cdot \frac{1}{2} = 1 \implies c = 2 \]

### Step 3: Write the Final Probability Function
Now that we have \( c \), we can write the probability function:
\[ P(X = n) = 2 \cdot \frac{1}{3^n} = \frac{2}{3^n} \]

### Step 4: Calculate the Expectation
The expectation \( E(X) \) is given by:
\[ E(X) = \sum_{n=1}^{\infty} n \cdot P(X = n) = \sum_{n=1}^{\infty} n \cdot \frac{2}{3^n} \]
To evaluate this sum, we can use the formula for the sum of a series involving \( n \):
\[ \sum_{n=1}^{\infty} n x^n = \frac{x}{(1-x)^2} \quad \text{for } |x| < 1 \]
Here, let \( x = \frac{1}{3} \):
\[ \sum_{n=1}^{\infty} n \left(\frac{1}{3}\right)^n = \frac{\frac{1}{3}}{\left(1 - \frac{1}{3}\right)^2} = \frac{\frac{1}{3}}{\left(\frac{2}{3}\right)^2} = \frac{\frac{1}{3}}{\frac{4}{9}} = \frac{3}{4} \]
Thus,
\[ E(X) = 2 \cdot \frac{3}{4} = \frac{3}{2} \]

### Final Answer
The expectation of \( X \) is:
\[ E(X) = \frac{3}{2} \]