The probability distribution of a random variable X is given by.
`{:(X=x:,0,1,2,3,4),(P(X=x):,0.4,0.3,0.1,0.1,0.1):}`
The variance of X, is

Options

A. 1.76
B. 2.45
C. 3.2
D. 4.8

Answer

To find the variance of the random variable \( X \) given its probability distribution, we can follow these steps:

### Step 1: Identify the values of \( X \) and their corresponding probabilities \( P(X=x) \)

From the given distribution:
- \( X = 0 \) with \( P(X=0) = 0.4 \)
- \( X = 1 \) with \( P(X=1) = 0.3 \)
- \( X = 2 \) with \( P(X=2) = 0.1 \)
- \( X = 3 \) with \( P(X=3) = 0.1 \)
- \( X = 4 \) with \( P(X=4) = 0.1 \)

### Step 2: Calculate \( E(X^2) \)

To find \( E(X^2) \), we need to calculate \( \sum (x^2 \cdot P(X=x)) \):

\[ E(X^2) = (0^2 \cdot 0.4) + (1^2 \cdot 0.3) + (2^2 \cdot 0.1) + (3^2 \cdot 0.1) + (4^2 \cdot 0.1) \]

Calculating each term:
- \( 0^2 \cdot 0.4 = 0 \)
- \( 1^2 \cdot 0.3 = 0.3 \)
- \( 2^2 \cdot 0.1 = 0.4 \)
- \( 3^2 \cdot 0.1 = 0.9 \)
- \( 4^2 \cdot 0.1 = 1.6 \)

Now sum these values:
\[ E(X^2) = 0 + 0.3 + 0.4 + 0.9 + 1.6 = 3.2 \]

### Step 3: Calculate \( E(X) \)

To find \( E(X) \), we need to calculate \( \sum (x \cdot P(X=x)) \):

\[ E(X) = (0 \cdot 0.4) + (1 \cdot 0.3) + (2 \cdot 0.1) + (3 \cdot 0.1) + (4 \cdot 0.1) \]

Calculating each term:
- \( 0 \cdot 0.4 = 0 \)
- \( 1 \cdot 0.3 = 0.3 \)
- \( 2 \cdot 0.1 = 0.2 \)
- \( 3 \cdot 0.1 = 0.3 \)
- \( 4 \cdot 0.1 = 0.4 \)

Now sum these values:
\[ E(X) = 0 + 0.3 + 0.2 + 0.3 + 0.4 = 1.2 \]

### Step 4: Calculate \( E(X)^2 \)

Now we need to square \( E(X) \):
\[ E(X)^2 = (1.2)^2 = 1.44 \]

### Step 5: Calculate the Variance

The variance \( Var(X) \) can be calculated using the formula:
\[ Var(X) = E(X^2) - E(X)^2 \]

Substituting the values we calculated:
\[ Var(X) = 3.2 - 1.44 = 1.76 \]

### Final Answer

The variance of \( X \) is \( 1.76 \).

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