The probability distribution of the random variable X is given in the following table. Find Var(X). (Round your answer to 4 decimal places.) 

   x

 1

 2

 3

 4

 5

   P(X =x)

 0.12

 0.24

 0.36

 0.17

 0.11



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

### Step 1: Calculate the expected value \( E(X) \).
The expected value is calculated as:
\[
E(X) = \sum (x_i \cdot P(X = x_i))
\]
where \( x_i \) are the possible values of \( X \) and \( P(X = x_i) \) are the corresponding probabilities.

From the table:
\[
E(X) = (1 \cdot 0.12) + (2 \cdot 0.24) + (3 \cdot 0.36) + (4 \cdot 0.17) + (5 \cdot 0.11)
\]
\[
E(X) = 0.12 + 0.48 + 1.08 + 0.68 + 0.55 = 2.91
\]

### Step 2: Calculate \( E(X^2) \).
To find \( E(X^2) \), we first square the \( x \)-values and then multiply by their respective probabilities:
\[
E(X^2) = \sum (x_i^2 \cdot P(X = x_i))
\]
\[
E(X^2) = (1^2 \cdot 0.12) + (2^2 \cdot 0.24) + (3^2 \cdot 0.36) + (4^2 \cdot 0.17) + (5^2 \cdot 0.11)
\]
\[
E(X^2) = (1 \cdot 0.12) + (4 \cdot 0.24) + (9 \cdot 0.36) + (16 \cdot 0.17) + (25 \cdot 0.11)
\]
\[
E(X^2) = 0.12 + 0.96 + 3.24 + 2.72 + 2.75 = 9.79
\]

### Step 3: Calculate the variance \( \text{Var}(X) \).
The variance is given by:
\[
\text{Var}(X) = E(X^2) - (E(X))^2
\]
\[
\text{Var}(X) = 9.79 - (2.91)^2 = 9.79 - 8.4681 = 1.3219
\]

So, the variance of \( X \) is \( \text{Var}(X) = 1.3219 \).

Would you like more details on any step, or do you have any questions?

### Relative questions:
1. How is the standard deviation related to variance?
2. What are the properties of a probability distribution?
3. How do you find the expected value of a continuous random variable?
4. What is the significance of variance in statistics?
5. Can you explain how the law of large numbers relates to expected value?

### Tip: 
Remember, variance measures how spread out the values of a random variable are from the mean.

The probability distribution of the random variable X is given in the following table. Find Var(X). (Round your answer to 4 decimal places.)

   x: 1, 2, 3, 4, 5
   P(X = x): 0.12, 0.24, 0.36, 0.17, 0.11

Learn how to calculate the variance of a discrete random variable given its probability distribution. This example involves calculating expected value (E(X)) and variance (Var(X)) step-by-step with detailed explanations. Suitable for high school students studying probability and statistics.

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