Final answer:
In this case, the value of P(y₁ > y₂ | y₁ < 2y₂) is 0.
None of the given options is correct
Step-by-step explanation:
To find the probability P(y₁ > y₂ | y₁< 2y₂), we can use the properties of exponential distributions.
Let's break down the steps:
1. Start by finding the distribution of the random variable Z = y₁ - y₂. Since y₁ and y₂ are independent, the difference Z will follow a Laplace distribution.
2. The Laplace distribution has a mean of 0 and a scale parameter of the sum of the scale parameters of y₁ and y₂, which is 14 in this case.
3. We want to calculate the probability P(y1 > y₂ | y₁ < 2y₂), which is the probability of Z > 0 and Z < y₁ - 2y₂.
4. However, calculating this probability directly can be complex. Instead, we can use the fact that the Laplace distribution is symmetric around its mean (0 in this case) to simplify the calculation.
5. Since the distribution is symmetric, P(Z > 0 and Z < y₁ - 2y₂) is equal to P(Z < 0 and Z > -y₁+ 2y₂).
6. Now we can rewrite the probability as: P(y₁ > y₂ | y₁ < 2y₂) = P(Z < 0 and Z > -y₁+ 2y₂) = P(Z > -y₁+ 2y₂) - P(Z > 0).
7. Since the Laplace distribution is continuous, P(Z > 0) is equal to 0.5.
8. Substitute the values into the equation: P(y₁ > y₂ | y₁ < 2y₂) = 0.5 - P(Z > -y₁ + 2y₂).
9. Finally, calculate P(Z > -y₁ + 2y₂) using the Laplace distribution. In this case, P(Z > -y1 + 2y₂) = 0.5e⁽⁻⁽⁻ʸ¹ ⁺ ²ʸ²⁾/⁽¹⁴/√²⁾⁾ = 0.5e⁽ʸ¹ ⁻ ²ʸ²⁾/⁽¹⁴/√².
10. Simplify the expression: P(Z > -y₁+ 2y₂) = 0.5e⁽ʸ¹ ⁻ ²ʸ²⁾/⁽¹⁴/√²⁾.
11. Substituting this back into the equation: P(y₁ > y₂ | y₁ < 2y₂) = 0.5 - 0.5e⁽ʸ¹ ⁻ ²ʸ²⁾/⁽¹⁴/√²⁾.
12. However, when we evaluate this equation, we find that it equals 0.
Therefore, the final answer is P(y₁ > y₂ | y₁ < 2y₂) = 0.
Therefore, the probability P(y₁ > y₂ | y₁ < 2y₂) cannot be expressed as a fraction and none of the options provided (A, B, C, D) are correct.