Let X be a random variable with the following probability distribution value exit X -3 is 0.30 -2 is 0.15 -1 is 0.30 and zero is 0.25 find the variance var(x) of x

To find the variance \( \text{Var}(X) \) of the random variable \( X \), we follow these steps:

1. **List the values of \( X \) and their corresponding probabilities:**

   \[
   \begin{array}{c|c}
   X & P(X) \\
   \hline
   -3 & 0.30 \\
   -2 & 0.15 \\
   -1 & 0.30 \\
   0 & 0.25 \\
   \end{array}
   \]

2. **Calculate the expected value \( \mathbb{E}(X) \):**

   \[
   \mathbb{E}(X) = \sum (X_i \cdot P(X_i)) 
   \]

   \[
   \mathbb{E}(X) = (-3)(0.30) + (-2)(0.15) + (-1)(0.30) + (0)(0.25) 
   \]

   \[
   \mathbb{E}(X) = -0.90 - 0.30 - 0.30 + 0 
   \]

   \[
   \mathbb{E}(X) = -1.50
   \]

3. **Calculate \( \mathbb{E}(X^2) \):**

   \[
   \mathbb{E}(X^2) = \sum (X_i^2 \cdot P(X_i)) 
   \]

   \[
   \mathbb{E}(X^2) = (-3)^2(0.30) + (-2)^2(0.15) + (-1)^2(0.30) + (0)^2(0.25) 
   \]

   \[
   \mathbb{E}(X^2) = 9(0.30) + 4(0.15) + 1(0.30) + 0 
   \]

   \[
   \mathbb{E}(X^2) = 2.70 + 0.60 + 0.30 
   \]

   \[
   \mathbb{E}(X^2) = 3.60
   \]

4. **Calculate the variance \( \text{Var}(X) \):**

   \[
   \text{Var}(X) = \mathbb{E}(X^2) - (\mathbb{E}(X))^2
   \]

   \[
   \text{Var}(X) = 3.60 - (-1.50)^2
   \]

   \[
   \text{Var}(X) = 3.60 - 2.25
   \]

   \[
   \text{Var}(X) = 1.35
   \]

So, the variance \( \text{Var}(X) \) is \( 1.35 \).

Would you like any further explanation or details on this problem? 

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**Relative Questions:**

1. How do you interpret the variance of a random variable?
2. What is the significance of the expected value \( \mathbb{E}(X) \)?
3. How does variance relate to standard deviation?
4. Can variance be negative? Why or why not?
5. How would the variance change if one of the probabilities in the distribution was increased?
6. What are some common applications of variance in real-world scenarios?
7. How is variance affected if all values of \( X \) are multiplied by a constant?
8. What happens to the variance if the distribution becomes more spread out?

**Tip:** When calculating the variance, remember that it measures the spread of the data around the mean, reflecting how much the values of the random variable deviate from the expected value.

Learn how to calculate the variance of a random variable with a specified probability distribution. Follow step-by-step calculations to find the variance using the expected value and second moment calculations. Suitable for undergraduate students and those interested in probability theory.

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