Question

In a locality 'A', the probability of a convective storm event is 0.7 with a density function \[ f_{X_1}(x_1) = e^{-x_1}, \; x_1 > 0 \] The probability of a tropical cyclone-induced storm in the same location is given by the density function \[ f_{X_2}(x_2) = 2 e^{-2x_2}, \; x_2 > 0 \] The probability of occurring more than 1 unit of storm event is \underline{\hspace{3cm}} (rounded off to 2 decimal places).

Hint:

When two storm types are possible, use the law of total probability: \(P = p_1P(A) + p_2P(B)\). For exponential, \(P(X>a) = e^{-\lambda a}\).

Answer 1

Step 1: Identify distribution. - For \(X_1\): exponential distribution with parameter \(\lambda = 1\). - For \(X_2\): exponential distribution with parameter \(\lambda = 2\).

Step 2: Probability of \(X_1 > 1\). For exponential distribution: \[ P(X > a) = e^{-\lambda a} \] So, \[ P(X_1 > 1) = e^{-1 \cdot 1} = e^{-1} = 0.3679 \]

Step 3: Probability of \(X_2 > 1\). \[ P(X_2 > 1) = e^{-2 \cdot 1} = e^{-2} = 0.1353 \]

Step 4: Total probability (law of total probability). \[ P(\text{storm} > 1) = 0.7 \cdot P(X_1 > 1) + 0.3 \cdot P(X_2 > 1) \] \[ = 0.7 \times 0.3679 + 0.3 \times 0.1353 \] \[ = 0.2575 + 0.0406 = 0.2981 \] Rounded to 2 decimal places: \[ \boxed{0.30} \]