Question

Consider a coin for which the probability of obtaining head in a single toss is \( \frac{1}{3} \). Sunita tosses the coin once. If head appears, she receives a random amount of \( X \) rupees, where \( X \) has the \( \text{Exp}\left( \frac{1}{9} \right) \) distribution. If tail appears, she loses a random amount of \( Y \) rupees, where \( Y \) has the \( \text{Exp}\left( \frac{1}{3} \right) \) distribution. Her expected gain (in rupees) is equal to __________ (answer in integer).

Answer 1

Correct Answer: 1

1. Expected Value of \( X \) and \( Y \): - For \( X \sim \text{Exp}\left(\frac{1}{9}\right) \), the mean is: \[ E(X) = 9. \] - For \( Y \sim \text{Exp}\left(\frac{1}{3}\right) \), the mean is: \[ E(Y) = 3. \] 2. Expected Gain: - The expected gain \( G \) is given by: \[ G = P(\text{Head})E(X) - P(\text{Tail})E(Y). \] - Substituting probabilities \( P(\text{Head}) = \frac{1}{3} \) and \( P(\text{Tail}) = \frac{2}{3} \), and the expected values: \[ G = \frac{1}{3}(9) - \frac{2}{3}(3). \] - Simplify: \[ G = 3 - 2 = 1. \]