exercise:B5799f68f3: Difference between revisions

Latest revision as of 17:48, 9 May 2023

An insurance company sells two types of auto insurance policies: Basic and Deluxe. The time until the next Basic Policy claim is an exponential random variable with mean two days. The time until the next Deluxe Policy claim is an independent exponential random variable with mean three days. Calculate the probability that the next claim will be a Deluxe Policy claim.

0.172
0.223
0.400
0.487
0.500

@@ Line 1: / Line 1: @@
-'''Solution: C'''
+An insurance company sells two types of auto insurance policies: Basic and Deluxe. The time until the next Basic Policy claim is an exponential random variable with mean two days. The time until the next Deluxe Policy claim is an independent exponential random variable with mean three days. Calculate the probability that the next claim will be a Deluxe Policy claim.
-Let <math>T_1</math> be the time until the next Basic Policy claim, and let <math>T_2</math> be the time until the next Deluxe policy claim. Then the joint pdf of <math>T_1</math> and <math>T_2</math> is
+<ul class="mw-excansopts">
+<li>0.172</li>
-<math display = "block">
+<li>0.223</li>
-f(t_1,t_2) = \left( \frac{1}{2} e^{-t_1/2}\right) \left( \frac{1}{3}e^{-t_2/3}\right) = \frac{1}{6}e^{-t_1/2}e^{-t_2/3}, 0 < t_1 < \infty, 0 < t_2 < \infty
+<li>0.400</li>
-</math>
+<li>0.487</li>
+<li>0.500</li>
-and we need to find
+</ul>
-<math display = "block">
-\begin{align*}
-P[T_1 < T_2] = \int_0^{\infty}\int_0^{t_1} \frac{1}{6} e^{-t_1/2}e^{-t_2/3} dt_2dt_1 &= \int_0^{\infty} \left[ -1\frac{1}{2} e^{-t_1/2}e^{-t_2/3}\right ]_0^{t_1} dt_1 \\
-&= \int_0^{\infty}[\frac{1}{2}e^{-t_1/2} - \frac{1}{2}e^{-t_1/2}e^{-t_1/3}] dt_1 \\
-&= \int_0^{\infty} \left[ \frac{1}{2}e^{-t_1/2} - \frac{1}{2} e^{-5t_1/6}\right] dt_1\\
-& = \left [ -e^{-t_1/2} + \frac{3}{5}e^{-5t_1/6}\right]_0^{\infty} \\
-&= 1- \frac{3}{5} = \frac{2}{5} \\
-& = 0.4
-\end{align*}
-</math>
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