Let's break down and solve the problem shown in the image. The question involves an exponential distribution with parameter $\beta = 4$.

(a) Completing the Sentence

The question asks about the distribution of $X$ and its expected value and standard deviation.

• The distribution of $X$ is exponential. In an exponential distribution, the rate parameter $\lambda = \frac{1}{\beta} = \frac{1}{4}$.
• The expected value $E(X)$ of an exponential distribution is equal to $\beta$, so $E(X) = 4$.
• The standard deviation $SD(X)$ is also $\beta$, so $SD(X) = 4$.

(b) Finding $P(2 \leq X \leq 5)$

For an exponential distribution, the cumulative distribution function (CDF) is: $F(x) = 1 - e^{-\lambda x}$ Where $\lambda = \frac{1}{4}$. To find $P(2 \leq X \leq 5)$, calculate: $P(2 \leq X \leq 5) = F(5) - F(2)$

First, calculate $F(5)$ and $F(2)$: $F(5) = 1 - e^{-\frac{1}{4} \cdot 5} = 1 - e^{-1.25}$ $F(2) = 1 - e^{-\frac{1}{4} \cdot 2} = 1 - e^{-0.5}$

So: $P(2 \leq X \leq 5) = (1 - e^{-1.25}) - (1 - e^{-0.5})$ You can use a calculator to find these values to four decimal places.

(c) Finding the 10th Percentile

To find the 10th percentile $X_{10}$, solve: $F(X_{10}) = 0.10$ This leads to the equation: $1 - e^{-\lambda X_{10}} = 0.10$ $e^{-\lambda X_{10}} = 0.90$ $-\lambda X_{10} = \ln(0.90)$ $X_{10} = -\frac{\ln(0.90)}{\lambda} = -\frac{\ln(0.90)}{\frac{1}{4}}$

This gives $X_{10}$. Solve using a calculator and round to two decimal places.

(d) Finding $P(X > 8.5)$

The probability that $X$ is greater than a certain value is: $P(X > 8.5) = 1 - F(8.5)$

First, calculate: $F(8.5) = 1 - e^{-\frac{1}{4} \cdot 8.5} = 1 - e^{-2.125}$ Then: $P(X > 8.5) = 1 - (1 - e^{-2.125}) = e^{-2.125}$ Calculate this to four decimal places.

Would you like detailed calculations for each part? Here are some related questions you might want to explore:

1. How do you find the mean and variance of an exponential distribution?
2. What is the significance of the percentile in statistics?
3. How do you compute probabilities for continuous distributions?
4. How does the exponential distribution relate to the Poisson process?
5. What are some real-world examples of exponential distributions?

Tip: The parameter $\lambda$ of the exponential distribution is the inverse of the average waiting time for an event to occur!