We are given the random variable $\mathrm{X}$ that follow an exponential distribution such as:
\[
X \sim \operatorname{Exp}\left(\frac{1}{9.848}\right)
\]
1. Find the expected value
2. Find the standard deviation
3. Find $P(X<12)$
4. Find $P(8
Solution
Step 1 ：The random variable X follows an exponential distribution, denoted as \(X \sim \operatorname{Exp}\left(\frac{1}{9.848}\right)\).
Step 2 ：The expected value of an exponential distribution is given by \(1/\lambda\), where \(\lambda\) is the rate parameter. In this case, \(\lambda = 1/9.848\). Therefore, the expected value is \(1/\lambda = 9.848\).
Step 3 ：The standard deviation of an exponential distribution is also given by \(1/\lambda\), so the standard deviation is also \(9.848\).
Step 4 ：To find \(P(X<12)\), we need to calculate the cumulative distribution function (CDF) at \(x=12\). The CDF of an exponential distribution is given by \(1 - e^{-\lambda x}\).
Step 5 ：To find \(P(8
Step 6 ：Final Answer: The expected value is \(\boxed{9.848}\). The standard deviation is \(\boxed{9.848}\). \(P(X<12) = \boxed{0.704}\). \(P(8
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Solution
Step 1 ：The random variable X follows an exponential distribution, denoted as \(X \sim \operatorname{Exp}\left(\frac{1}{9.848}\right)\).
Step 2 ：The expected value of an exponential distribution is given by \(1/\lambda\), where \(\lambda\) is the rate parameter. In this case, \(\lambda = 1/9.848\). Therefore, the expected value is \(1/\lambda = 9.848\).
Step 3 ：The standard deviation of an exponential distribution is also given by \(1/\lambda\), so the standard deviation is also \(9.848\).
Step 4 ：To find \(P(X<12)\), we need to calculate the cumulative distribution function (CDF) at \(x=12\). The CDF of an exponential distribution is given by \(1 - e^{-\lambda x}\).
Step 5 ：To find \(P(8 Step 6 ：Final Answer: The expected value is \(\boxed{9.848}\). The standard deviation is \(\boxed{9.848}\). \(P(X<12) = \boxed{0.704}\). \(P(8
