If X is the a random variable with probability mass function `P(x) = kx `, for x = 1,2,3
= 0 , otherwise
then k = ……..

Options

A. `(1)/(5)`
B. `(1)/(4)`
C. `(1)/(6)`
D. `(2)/(3)`

Answer

To find the value of \( k \) in the probability mass function \( P(x) = kx \) for \( x = 1, 2, 3 \), we need to ensure that the total probability sums up to 1. Here’s how we can solve it step by step:

### Step 1: Write down the probability mass function
The probability mass function is given as:
\[ P(x) = kx \quad \text{for } x = 1, 2, 3 \]
\[ P(x) = 0 \quad \text{otherwise} \]

### Step 2: Write the probabilities for each value of \( x \)
We can calculate the probabilities for \( x = 1, 2, 3 \):
- For \( x = 1 \):
\[ P(1) = k \cdot 1 = k \]
- For \( x = 2 \):
\[ P(2) = k \cdot 2 = 2k \]
- For \( x = 3 \):
\[ P(3) = k \cdot 3 = 3k \]

### Step 3: Set up the equation for total probability
The total probability must equal 1:
\[ P(1) + P(2) + P(3) = 1 \]
Substituting the values we calculated:
\[ k + 2k + 3k = 1 \]

### Step 4: Simplify the equation
Combine like terms:
\[ 6k = 1 \]

### Step 5: Solve for \( k \)
Now, divide both sides by 6:
\[ k = \frac{1}{6} \]

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{\frac{1}{6}} \]