https://file.aiquickdraw.com/mathbot/file/1730237952245xhouhykz.jpg

also the options for the distribution of x is skewed to the left, right or symmetrical

Let’s go through each part of this problem one by one.

### Given Data
The random variable \( X \) follows an exponential distribution with \( \beta = 3 \).

For an exponential distribution:
- The mean \( E(X) = \beta \)
- The standard deviation \( SD(X) = \beta \)
- The probability density function is \( f(x) = \frac{1}{\beta} e^{-x / \beta} \) for \( x \geq 0 \).

### Solution

---

#### (a) Complete the sentence:
The distribution of \( X \) is **skewed to the right** (this is a property of the exponential distribution, which is right-skewed).

For the exponential distribution with \( \beta = 3 \):
- \( E(X) = \beta = 3 \)
- \( SD(X) = \beta = 3 \)

So, we have:
- **Answer**: The distribution of \( X \) is skewed to the right, with \( E(X) = 3 \) and \( SD(X) = 3 \).

---

#### (b) Find \( P(1 \leq X \leq 4) \)
For an exponential distribution, the cumulative distribution function (CDF) is given by:
\[
P(X \leq x) = 1 - e^{-x / \beta}
\]

To find \( P(1 \leq X \leq 4) \), we calculate:
\[
P(1 \leq X \leq 4) = P(X \leq 4) - P(X \leq 1)
\]

Using \( \beta = 3 \):
1. \( P(X \leq 4) = 1 - e^{-4 / 3} \)
2. \( P(X \leq 1) = 1 - e^{-1 / 3} \)

Calculating these values:
- \( P(X \leq 4) \approx 1 - e^{-1.3333} \approx 1 - 0.2636 = 0.7364 \)
- \( P(X \leq 1) \approx 1 - e^{-0.3333} \approx 1 - 0.7173 = 0.2827 \)

Thus,
\[
P(1 \leq X \leq 4) \approx 0.7364 - 0.2827 = 0.4537
\]

**Answer**: \( P(1 \leq X \leq 4) \approx 0.4537 \)

---

#### (c) Find the 25th percentile of \( X \)
The \( p \)-th percentile for an exponential distribution is given by:
\[
X_p = -\beta \ln(1 - p)
\]

For the 25th percentile, \( p = 0.25 \) and \( \beta = 3 \):
\[
X_{25} = -3 \ln(1 - 0.25) = -3 \ln(0.75)
\]

Calculating this:
\[
X_{25} \approx -3 \times (-0.2877) = 0.8631
\]

**Answer**: \( X_{25} \approx 0.86 \)

---

#### (d) Find \( P(X > 9.5) \)
The probability \( P(X > x) \) for an exponential distribution is:
\[
P(X > x) = e^{-x / \beta}
\]

Using \( x = 9.5 \) and \( \beta = 3 \):
\[
P(X > 9.5) = e^{-9.5 / 3} = e^{-3.1667}
\]

Calculating this:
\[
P(X > 9.5) \approx 0.0424
\]

**Answer**: \( P(X > 9.5) \approx 0.0424 \)

---

### Summary of Answers
1. **(a)** The distribution of \( X \) is skewed to the right, with \( E(X) = 3 \) and \( SD(X) = 3 \).
2. **(b)** \( P(1 \leq X \leq 4) \approx 0.4537 \)
3. **(c)** \( X_{25} \approx 0.86 \)
4. **(d)** \( P(X > 9.5) \approx 0.0424 \)

Would you like further details on any of these parts, or have any questions?

---

### Additional Questions
1. How would the results change if \( \beta \) was different?
2. Can the exponential distribution ever be symmetric?
3. What is the median of an exponential distribution with \( \beta = 3 \)?
4. How do we interpret the 25th percentile in terms of the distribution?
5. Why is the exponential distribution often used to model waiting times?

### Tip
For exponential distributions, the mean and standard deviation are always equal to \( \beta \), which simplifies calculations when describing the spread of data.

A random variable X can be modeled by the exponential distribution with β = 3. 
(a) Complete the following sentence by selecting the correct skewness of the distribution. Also, find the mean and standard deviation.
(b) Find P(1 ≤ X ≤ 4) and round to four decimals.
(c) Find the 25th percentile of X, using two decimals in the answer.
(d) Find P(X > 9.5) and round to four decimals.

Explore the properties of an exponential distribution with β = 3, including how to find the mean, standard deviation, and skewness. Solve for probabilities P(1 ≤ X ≤ 4) and P(X > 9.5), and calculate the 25th percentile. Perfect for high school students studying probability distributions.

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