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Solve this question fully mathematically and carefully.

To solve this problem, we are given a probability density function (PDF) \( f(x) \) defined as:

\[
f(x) = \begin{cases} 
\frac{1}{4} e^{-x/4} & \text{for } x \geq 0 \\
0 & \text{elsewhere}
\end{cases}
\]

This describes the length of time \( Y \) in the air for an airplane to obtain clearance for takeoff, where \( Y = 3X + 2 \).

Let's solve the required parts step by step:

### a) Mean length \( E(y) \)
The mean of the random variable \( Y = 3X + 2 \) can be found using the linearity of expectation:
\[
E(Y) = E(3X + 2) = 3E(X) + 2
\]
First, let's find \( E(X) \). For an exponential distribution \( f(x) = \frac{1}{\lambda} e^{-x/\lambda} \) where \( \lambda = 4 \):

\[
E(X) = \lambda = 4
\]

So,
\[
E(Y) = 3(4) + 2 = 12 + 2 = 14
\]

### b) Variance \( \sigma^2(y) \)
The variance of \( Y = 3X + 2 \) is calculated as:

\[
\text{Var}(Y) = \text{Var}(3X + 2) = 3^2 \text{Var}(X) = 9 \text{Var}(X)
\]
For an exponential distribution, the variance is:
\[
\text{Var}(X) = \lambda^2 = 16
\]

Therefore,
\[
\text{Var}(Y) = 9 \times 16 = 144
\]

### c) Standard Deviation \( \sigma(y) \)
The standard deviation is the square root of the variance:

\[
\sigma(Y) = \sqrt{144} = 12
\]

### d) Probability \( P(Y \leq 1.5) \)
To find \( P(Y \leq 1.5) \), we first need to convert this into a probability involving \( X \):

Given \( Y = 3X + 2 \), we solve for \( X \) when \( Y = 1.5 \):

\[
1.5 = 3X + 2 \quad \Rightarrow \quad 3X = 1.5 - 2 = -0.5 \quad \Rightarrow \quad X = -\frac{1}{6}
\]

Since \( X \geq 0 \) in the PDF, \( P(Y \leq 1.5) = 0 \) because the value of \( X = -\frac{1}{6} \) is not in the domain of the distribution.

So,
\[
P(Y \leq 1.5) = 0
\]

### Summary of Results:
1. **Mean** \( E(Y) = 14 \)
2. **Variance** \( \text{Var}(Y) = 144 \)
3. **Standard Deviation** \( \sigma(Y) = 12 \)
4. **Probability** \( P(Y \leq 1.5) = 0 \)

Do you want more details or have any questions?

Here are some related questions:
1. How would the mean change if \( Y = 5X + 3 \)?
2. What would be the probability \( P(Y > 20) \)?
3. How would the variance be affected if \( Y = 2X + 1 \)?
4. What is the cumulative distribution function (CDF) for \( Y \)?
5. Can you find the mode of \( Y \) given the exponential distribution of \( X \)?

**Tip:** Remember that the linear transformation of a random variable affects both the mean and variance, but only scaling by a factor (not adding constants) affects the variance.

The length of time in air for an airplane to obtain clearance takeoff at a certain airport is a random variable Y = 3X + 2 where X has a density function f(x) = (1/4)e^(-x/4) for x ≥ 0, and 0 elsewhere. Determine the mean length E(Y), variance σ²(Y), standard deviation σ(Y), and P(Y ≤ 1.5).

Learn how to solve probability density function problems involving exponential distributions and variable transformations. This problem covers calculating the mean, variance, standard deviation, and probability of a transformed random variable. Suitable for undergraduate students.

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