Find the mean, variance and quartiles 

*Q*1, *Q*3

 of the following random variables:
(1) U is a random variable with uniform density over [-2,18]
Mean =   , Variance =   , 


*Q*1= 

   , 

*Q*3= 

  
(2) V is a random variable with uniform density over [-2,25]
Mean =   , Variance =   , 


*Q*1= 

   , 

*Q*3= 

  



To find the mean, variance, and quartiles \(Q_1\) and \(Q_3\) of a uniform random variable, we can use the properties of the uniform distribution. For a uniform random variable \(X\) over the interval \([a, b]\):

1. **Mean**: 
   \[ \mu = \frac{a + b}{2} \]

2. **Variance**: 
   \[ \sigma^2 = \frac{(b - a)^2}{12} \]

3. **First Quartile \(Q_1\)**:
   \[ Q_1 = a + \frac{1}{4}(b - a) \]

4. **Third Quartile \(Q_3\)**:
   \[ Q_3 = a + \frac{3}{4}(b - a) \]

Let's apply these formulas to each random variable.

### 1. \(U\) is a random variable with uniform density over \([-2, 18]\)

- **Mean**: 
  \[ \mu = \frac{-2 + 18}{2} = \frac{16}{2} = 8 \]

- **Variance**: 
  \[ \sigma^2 = \frac{(18 - (-2))^2}{12} = \frac{(20)^2}{12} = \frac{400}{12} = \frac{100}{3} \approx 33.33 \]

- **First Quartile \(Q_1\)**:
  \[ Q_1 = -2 + \frac{1}{4}(18 - (-2)) = -2 + \frac{1}{4}(20) = -2 + 5 = 3 \]

- **Third Quartile \(Q_3\)**:
  \[ Q_3 = -2 + \frac{3}{4}(18 - (-2)) = -2 + \frac{3}{4}(20) = -2 + 15 = 13 \]

### 2. \(V\) is a random variable with uniform density over \([-2, 25]\)

- **Mean**: 
  \[ \mu = \frac{-2 + 25}{2} = \frac{23}{2} = 11.5 \]

- **Variance**: 
  \[ \sigma^2 = \frac{(25 - (-2))^2}{12} = \frac{(27)^2}{12} = \frac{729}{12} = 60.75 \]

- **First Quartile \(Q_1\)**:
  \[ Q_1 = -2 + \frac{1}{4}(25 - (-2)) = -2 + \frac{1}{4}(27) = -2 + 6.75 = 4.75 \]

- **Third Quartile \(Q_3\)**:
  \[ Q_3 = -2 + \frac{3}{4}(25 - (-2)) = -2 + \frac{3}{4}(27) = -2 + 20.25 = 18.25 \]

### Summary:

1. For \( U \) (uniform density over \([-2, 18]\)):
   - Mean = \( 8 \)
   - Variance = \( \frac{100}{3} \approx 33.33 \)
   - \( Q_1 = 3 \)
   - \( Q_3 = 13 \)

2. For \( V \) (uniform density over \([-2, 25]\)):
   - Mean = \( 11.5 \)
   - Variance = \( 60.75 \)
   - \( Q_1 = 4.75 \)
   - \( Q_3 = 18.25 \)

Would you like more details or have any questions? Here are some related questions to consider:

1. What is the standard deviation for the random variables \( U \) and \( V \)?
2. How would these values change if the interval for \( U \) was \([-3, 19]\)?
3. What is the interquartile range (IQR) for \( U \) and \( V \)?
4. How can we use these quartiles to identify outliers in a dataset?
5. What is the cumulative distribution function (CDF) for \( U \) and \( V \)?
6. How do these properties compare for a normal distribution with the same mean and variance?
7. What are the skewness and kurtosis of a uniform distribution?
8. How can we simulate random samples from these uniform distributions?

**Tip:** Understanding the properties of different probability distributions, such as mean and variance, can help in comparing and contrasting their behaviors and applications in real-world scenarios.

Learn how to calculate the mean, variance, first quartile (Q1), and third quartile (Q3) of uniform random variables with specific intervals. This problem involves mathematical concepts of uniform distribution, mean, variance, and quartiles. Suitable for advanced high school students and above.

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