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: $$G = 3 - 2 = 1.$$