Let X be the lifetime of a microwave oven. Assume that X follows a uniform distribution between 6 and 14 years. 

a. Calculate the mean of this distribution. 
b. Calculate the standard deviation of this distribution. 
c. What is the probability a particular microwave oven lasts between 10 and 14 years? 
d. What is the probability a microwave oven will last less than 8 years?

Let \( X \) be the lifetime of a microwave oven, which follows a uniform distribution between 6 and 14 years. A uniform distribution over an interval \([a, b]\) is denoted as \( X \sim U(a, b) \).

### Given:
- \( a = 6 \) (minimum value)
- \( b = 14 \) (maximum value)

### Part (a): Mean of the distribution
The mean \( \mu \) of a uniform distribution \( U(a, b) \) is given by:

\[
\mu = \frac{a + b}{2}
\]

\[
\mu = \frac{6 + 14}{2} = 10
\]

### Answer to (a):
The mean of this distribution is **10 years**.

---

### Part (b): Standard deviation of the distribution
The standard deviation \( \sigma \) of a uniform distribution \( U(a, b) \) is given by:

\[
\sigma = \frac{b - a}{\sqrt{12}}
\]

\[
\sigma = \frac{14 - 6}{\sqrt{12}} = \frac{8}{\sqrt{12}} \approx 2.31
\]

### Answer to (b):
The standard deviation of this distribution is approximately **2.31 years**.

---

### Part (c): Probability that a microwave lasts between 10 and 14 years
The probability density function (PDF) of a uniform distribution \( U(a, b) \) is given by:

\[
f(x) = \frac{1}{b - a}
\]

The probability that \( X \) lies between two values \( x_1 \) and \( x_2 \) within the range \([a, b]\) is given by:

\[
P(x_1 \leq X \leq x_2) = \frac{x_2 - x_1}{b - a}
\]

For \( x_1 = 10 \) and \( x_2 = 14 \):

\[
P(10 \leq X \leq 14) = \frac{14 - 10}{14 - 6} = \frac{4}{8} = 0.5
\]

### Answer to (c):
The probability that a microwave oven lasts between 10 and 14 years is **0.5**.

---

### Part (d): Probability that a microwave lasts less than 8 years
Using the same formula:

\[
P(X < 8) = \frac{8 - a}{b - a} = \frac{8 - 6}{14 - 6} = \frac{2}{8} = 0.25
\]

### Answer to (d):
The probability that a microwave oven lasts less than 8 years is **0.25**.

---

Would you like a more detailed explanation of any part of this solution?

### Related Questions:
1. What is the probability that a microwave oven lasts more than 12 years?
2. How does the uniform distribution differ from other continuous distributions like the normal distribution?
3. What is the cumulative distribution function (CDF) for a uniform distribution, and how can it be used here?
4. How do we find the median for a uniform distribution between 6 and 14 years?
5. How does changing the interval [6, 14] affect the mean and standard deviation of the distribution?

#### Tip:
The mean of a uniform distribution is always the midpoint of the interval \([a, b]\). This makes it easy to calculate without needing any complex formulas!

Let X be the lifetime of a microwave oven. Assume that X follows a uniform distribution between 6 and 14 years.

a. Calculate the mean of this distribution.
b. Calculate the standard deviation of this distribution.
c. What is the probability a particular microwave oven lasts between 10 and 14 years?
d. What is the probability a microwave oven will last less than 8 years?

Learn how to calculate the mean and standard deviation of a uniform distribution between 6 and 14 years, as well as the probabilities of a microwave lasting between 10 and 14 years, and less than 8 years. This example covers key concepts in probability and statistics, suitable for college-level studies.

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