Question

If the p.m.f. is given by \( P(X) = k \left( \frac{4}{x} \right) \) for \( x = 0, 1, 2, 3, 4 \) and \( k>0 \), then the value of \( k \) is

• A. \( \frac{3}{16} \)
• B. \( \frac{7}{16} \)
• C. \( \frac{1}{16} \)
• D. \( \frac{5}{16} \)

Hint:

For p.m.f. calculations, remember to ensure that the sum of the probabilities equals 1 and use this constraint to find any unknown constants.

Answer 1

The Correct Option is C

Step 1: Use the condition for the sum of probabilities.
For a probability mass function, the sum of all probabilities must equal 1: \[ \sum_{x=0}^{4} P(X) = 1 \]
Step 2: Set up the equation.
Substitute the given p.m.f. into the sum: \[ \sum_{x=0}^{4} k \left( \frac{4}{x} \right) = 1 \]
Step 3: Solve for \( k \).
Now, solve for \( k \) by simplifying the equation: \[ k \left( \frac{4}{x} \right) = \frac{1}{16} \] The correct value of \( k \) is \( \frac{1}{16} \).