 Find the mean, variance, and standard deviation of the random variable X
 associated with the probability density function over the indicated interval.
f(x)=6/64   (x−2)(6−x);2≤x≤6 


To find the **mean**, **variance**, and **standard deviation** of the random variable \( X \) associated with the probability density function (pdf) \( f(x) = \frac{6}{64}(x-2)(6-x) \) over the interval \( 2 \leq x \leq 6 \), we follow these steps:

### Step 1: Normalizing the Probability Density Function
First, verify that the given function is a valid pdf by ensuring the total probability over the interval is 1. This is done by integrating \( f(x) \) over the given interval:
\[
\int_{2}^{6} f(x) \, dx = 1
\]
where \( f(x) = \frac{6}{64}(x-2)(6-x) \).

### Step 2: Mean (Expected Value)
The **mean** or **expected value** \( \mu \) of the random variable \( X \) is given by:
\[
\mu = E[X] = \int_{2}^{6} x f(x) \, dx
\]

### Step 3: Variance
The **variance** \( \sigma^2 \) is given by:
\[
\sigma^2 = E[X^2] - (E[X])^2
\]
where \( E[X^2] \) is the expected value of \( X^2 \):
\[
E[X^2] = \int_{2}^{6} x^2 f(x) \, dx
\]

### Step 4: Standard Deviation
The **standard deviation** \( \sigma \) is simply the square root of the variance:
\[
\sigma = \sqrt{\sigma^2}
\]

Now let's compute these step by step.

#### 1. Normalizing the pdf:
First, we check if the total probability integrates to 1:
\[
\int_{2}^{6} \frac{6}{64}(x-2)(6-x) \, dx
\]

#### 2. Mean (Expected Value):
We compute the mean by evaluating:
\[
\mu = \int_{2}^{6} x \frac{6}{64}(x-2)(6-x) \, dx
\]

#### 3. Variance:
We compute \( E[X^2] \) by evaluating:
\[
E[X^2] = \int_{2}^{6} x^2 \frac{6}{64}(x-2)(6-x) \, dx
\]
Then we calculate the variance using \( \sigma^2 = E[X^2] - (E[X])^2 \).

I'll compute these integrals to get the exact values.Here are the computed results for the random variable \( X \) with the given probability density function:

1. **Normalization**: The total probability integrates to 1, confirming that \( f(x) \) is a valid probability density function.
   
2. **Mean (Expected Value)**: The mean is \( \mu = 4 \).

3. **Variance**: The variance is \( \sigma^2 = 0.8 \).

4. **Standard Deviation**: The standard deviation is \( \sigma \approx 0.894 \).

Would you like more details on any of these calculations or have further questions?

Here are 5 related questions you might explore:
1. How do we compute the mode of a continuous random variable?
2. What does the shape of this probability distribution look like?
3. How does variance relate to the spread of data in a probability distribution?
4. Can we compute higher moments of this distribution (e.g., skewness, kurtosis)?
5. How does the interval \( [2, 6] \) affect the mean and variance?

**Tip**: The standard deviation gives insight into the spread of values around the mean—lower values indicate that data points are closely clustered around the mean.

Find the mean, variance, and standard deviation of the random variable X associated with the probability density function over the indicated interval. f(x)=6/64 (x−2)(6−x); 2≤x≤6.

Discover how to calculate the mean, variance, and standard deviation for the random variable X with a probability density function f(x) = 6/64 (x−2)(6−x) over the interval [2,6]. This detailed explanation covers key concepts in probability theory, including expected value and variance, ideal for students at an undergraduate level.

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