Question

If \( P(X = x) = k \left( \frac{3}{8} \right)^x \), where \( x = 1, 2, 3, \dots \) is the probability distribution function of a discrete random variable X, then \( k = \)

• A. \( \frac{5}{8} \)
• B. \( \frac{8}{3} \)
• C. \( \frac{5}{3} \)
• D. \( \frac{4}{3} \)

Hint:

For a probability distribution, ensure that the sum of the probabilities equals 1. Use the sum of a geometric series to calculate the value of \( k \).

Answer 1

The Correct Option is C

We are given the probability distribution function: \[ P(X = x) = k \left( \frac{3}{8} \right)^x \quad \text{for} \quad x = 1, 2, 3, \dots \] For \( P(X) \) to be a valid probability distribution, the sum of all probabilities must be 1, i.e., \[ \sum_{x=1}^{\infty} P(X = x) = 1 \] Substitute the given expression for \( P(X = x) \): \[ \sum_{x=1}^{\infty} k \left( \frac{3}{8} \right)^x = 1 \] Factor out \( k \) from the sum: \[ k \sum_{x=1}^{\infty} \left( \frac{3}{8} \right)^x = 1 \] The sum is a geometric series with the first term \( a = \left( \frac{3}{8} \right) \) and the common ratio \( r = \frac{3}{8} \). The sum of an infinite geometric series is given by: \[ \sum_{x=1}^{\infty} r^x = \frac{r}{1 - r} \quad \text{for} \quad |r|<1 \] Substitute \( r = \frac{3}{8} \): \[ \sum_{x=1}^{\infty} \left( \frac{3}{8} \right)^x = \frac{\frac{3}{8}}{1 - \frac{3}{8}} = \frac{\frac{3}{8}}{\frac{5}{8}} = \frac{3}{5} \] Thus, the equation becomes: \[ k \times \frac{3}{5} = 1 \] Solve for \( k \): \[ k = \frac{5}{3} \] Thus, the correct answer is option (3), \( \frac{5}{3} \).

Answer 2

Given:
The probability mass function (PMF) of a discrete random variable \( X \) is:
\( P(X = x) = k \left( \frac{3}{8} \right)^x \), where \( x = 1, 2, 3, \dots \)

Step 1: Condition for a valid PMF
For any valid probability distribution function, the total probability must be 1:
\[ \sum_{x=1}^{\infty} P(X = x) = 1 \]
Substitute the given PMF:
\[ \sum_{x=1}^{\infty} k \left( \frac{3}{8} \right)^x = 1 \]
Factor out the constant \( k \):
\[ k \sum_{x=1}^{\infty} \left( \frac{3}{8} \right)^x = 1 \]

Step 2: Use geometric series formula
We use the infinite geometric series formula:
\[ \sum_{x=1}^{\infty} r^x = \frac{r}{1 - r} \quad \text{for } |r| < 1 \]
Here, \( r = \frac{3}{8} \), which is less than 1.
So,
\[ \sum_{x=1}^{\infty} \left( \frac{3}{8} \right)^x = \frac{\frac{3}{8}}{1 - \frac{3}{8}} = \frac{\frac{3}{8}}{\frac{5}{8}} = \frac{3}{5} \]

Step 3: Solve for \( k \)
Now substitute back:
\[ k \cdot \frac{3}{5} = 1 \Rightarrow k = \frac{5}{3} \]

Final Answer:
\( \boxed{\frac{5}{3}} \)
Therefore, the value of \( k \) that makes the given function a valid probability distribution is \( \frac{5}{3} \).