A random variable X takes values `0, 1, 2,......` with probability proportional to (x with proba bility proportional to `(x + 1)(1/5)^x`, , then `5 * [P(x leq 1)^(1/2)]` equals

Options

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

Answer

To solve the problem step by step, we need to find the value of \( 5 \times [P(X \leq 1)]^{1/2} \) given that the probability \( P(X) \) is proportional to \( (X + 1) \left( \frac{1}{5} \right)^X \).

### Step 1: Define the Probability Function
Since \( P(X) \) is proportional to \( (X + 1) \left( \frac{1}{5} \right)^X \), we can express it as:
\[ P(X) = k \cdot (X + 1) \left( \frac{1}{5} \right)^X \]
where \( k \) is a normalization constant.

### Step 2: Calculate Probabilities for \( X = 0, 1, 2 \)
Now, we will calculate the probabilities for \( X = 0, 1, 2 \).

- For \( X = 0 \):
\[ P(0) = k \cdot (0 + 1) \left( \frac{1}{5} \right)^0 = k \cdot 1 = k \]

- For \( X = 1 \):
\[ P(1) = k \cdot (1 + 1) \left( \frac{1}{5} \right)^1 = k \cdot 2 \cdot \frac{1}{5} = \frac{2k}{5} \]

- For \( X = 2 \):
\[ P(2) = k \cdot (2 + 1) \left( \frac{1}{5} \right)^2 = k \cdot 3 \cdot \frac{1}{25} = \frac{3k}{25} \]

### Step 3: Set Up the Normalization Condition
The sum of all probabilities must equal 1:
\[ P(0) + P(1) + P(2) = 1 \]
Substituting the values we calculated:
\[ k + \frac{2k}{5} + \frac{3k}{25} = 1 \]

### Step 4: Find a Common Denominator
The common denominator for the fractions is 25. Rewriting the equation:
\[ 25k + 10k + 3k = 25 \]
\[ 38k = 25 \]

### Step 5: Solve for \( k \)
Now, we can solve for \( k \):
\[ k = \frac{25}{38} \]

### Step 6: Calculate \( P(X \leq 1) \)
Now we need to find \( P(X \leq 1) \):
\[ P(X \leq 1) = P(0) + P(1) = k + \frac{2k}{5} \]
Substituting \( k \):
\[ P(X \leq 1) = \frac{25}{38} + \frac{2 \cdot \frac{25}{38}}{5} = \frac{25}{38} + \frac{10}{38} = \frac{35}{38} \]

### Step 7: Calculate \( 5 \times [P(X \leq 1)]^{1/2} \)
Now we can calculate:
\[ 5 \times [P(X \leq 1)]^{1/2} = 5 \times \left(\frac{35}{38}\right)^{1/2} \]
Calculating \( \left(\frac{35}{38}\right)^{1/2} \):
\[ \sqrt{\frac{35}{38}} = \frac{\sqrt{35}}{\sqrt{38}} \]
Thus,
\[ 5 \times \sqrt{\frac{35}{38}} = \frac{5\sqrt{35}}{\sqrt{38}} \]

### Final Result
To simplify, we can approximate or leave it as is, but the answer in the context of the question is:
\[ \text{Final Answer} = 5 \times \left(\frac{35}{38}\right)^{1/2} \]