If the range ot a random vaniabie `X` is `0,1,2,3,` at `P(X=K)=((K+1)/3^k)` a for `k>=0,` then a equals

Options

A. `(2)/(3)`
B. `(4)/(9)`
C. `(8)/(27)`
D. (16)/(81)`

Answer

To solve the problem, we need to find the value of \( a \) in the probability function given for the random variable \( X \).

### Step-by-Step Solution:

1. **Understanding the Probability Function**:
The probability function is given as:
\[ P(X = k) = \frac{k + 1}{3^k} \cdot a \quad \text{for } k \geq 0 \]
The range of \( X \) is \( 0, 1, 2, 3, \ldots \).

2. **Using the Property of Probability**:
The sum of all probabilities must equal 1:
\[ \sum_{k=0}^{\infty} P(X = k) = 1 \]
Therefore, we can write:
\[ \sum_{k=0}^{\infty} \left( \frac{k + 1}{3^k} \cdot a \right) = 1 \]

3. **Factoring Out \( a \)**:
Since \( a \) is a constant, we can factor it out of the summation:
\[ a \cdot \sum_{k=0}^{\infty} \frac{k + 1}{3^k} = 1 \]

4. **Splitting the Summation**:
We can split the summation into two parts:
\[ \sum_{k=0}^{\infty} \frac{k + 1}{3^k} = \sum_{k=0}^{\infty} \frac{k}{3^k} + \sum_{k=0}^{\infty} \frac{1}{3^k} \]

5. **Calculating the Second Summation**:
The second summation is a geometric series:
\[ \sum_{k=0}^{\infty} \frac{1}{3^k} = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2} \]

6. **Calculating the First Summation**:
To find \( \sum_{k=0}^{\infty} \frac{k}{3^k} \), we can use the formula for the sum of an infinite series:
\[ S = \sum_{k=0}^{\infty} kx^k = \frac{x}{(1-x)^2} \quad \text{for } |x| < 1 \]
Here, \( x = \frac{1}{3} \):
\[ \sum_{k=0}^{\infty} k \left( \frac{1}{3} \right)^k = \frac{\frac{1}{3}}{\left(1 - \frac{1}{3}\right)^2} = \frac{\frac{1}{3}}{\left(\frac{2}{3}\right)^2} = \frac{\frac{1}{3}}{\frac{4}{9}} = \frac{3}{4} \]

7. **Combining the Results**:
Now we can combine the results of the two summations:
\[ \sum_{k=0}^{\infty} \frac{k + 1}{3^k} = \sum_{k=0}^{\infty} \frac{k}{3^k} + \sum_{k=0}^{\infty} \frac{1}{3^k} = \frac{3}{4} + \frac{3}{2} = \frac{3}{4} + \frac{6}{4} = \frac{9}{4} \]

8. **Substituting Back**:
Now substituting back into the equation:
\[ a \cdot \frac{9}{4} = 1 \]
Thus,
\[ a = \frac{4}{9} \]

### Final Answer:
The value of \( a \) is \( \frac{4}{9} \).