A random variable has the following distribution.
`{:(x=x_(i),,-3,,-2,,-1,,0,,1,,2),(P(x=x_(i)),,0.1,,2K,,3K,,7K,,0.2,,0.1):}`
Then for the values, A = K, B = Mean, C = Variance, the ascending order is

Options

A. a) A, B, C
B. b) B, A, C
C. c) C, B, A
D. d) C, A, B

Answer

To solve the problem step by step, we need to find the values of \( K \), the mean \( B \), and the variance \( C \) of the given random variable distribution. The ascending order of these values will be determined at the end.

### Step 1: Finding \( K \)

We know that the sum of all probabilities must equal 1. The given probabilities are:

- \( P(X = -3) = 0.1 \)
- \( P(X = -2) = 2K \)
- \( P(X = -1) = 3K \)
- \( P(X = 0) = 7K \)
- \( P(X = 1) = 0.2 \)
- \( P(X = 2) = 0.1 \)

Setting up the equation:

\[ 0.1 + 2K + 3K + 7K + 0.2 + 0.1 = 1 \]

Combining like terms:

\[ 0.4 + 12K = 1 \]

Solving for \( K \):

\[ 12K = 1 - 0.4 \]
\[ 12K = 0.6 \]
\[ K = \frac{0.6}{12} = 0.05 \]

Thus, \( A = K = 0.05 \).

### Step 2: Finding the Mean \( B \)

The mean \( B \) (or \( \mu \)) is calculated using the formula:

\[ \mu = \sum (x_i \cdot P(x_i)) \]

Calculating each term:

\[ \mu = (-3 \cdot 0.1) + (-2 \cdot 2K) + (-1 \cdot 3K) + (0 \cdot 7K) + (1 \cdot 0.2) + (2 \cdot 0.1) \]

Substituting \( K = 0.05 \):

\[ \mu = (-3 \cdot 0.1) + (-2 \cdot 0.1) + (-1 \cdot 0.15) + (0) + (0.2) + (0.2) \]

Calculating each term:

\[ \mu = -0.3 - 0.1 - 0.15 + 0 + 0.2 + 0.2 \]
\[ \mu = -0.3 - 0.1 - 0.15 + 0.4 \]
\[ \mu = -0.25 \]

Thus, \( B = \mu = -0.25 \).

### Step 3: Finding the Variance \( C \)

The variance \( C \) is calculated using the formula:

\[ C = \sum (x_i^2 \cdot P(x_i)) - \mu^2 \]

Calculating \( \sum (x_i^2 \cdot P(x_i)) \):

\[ C = [(-3)^2 \cdot 0.1] + [(-2)^2 \cdot 2K] + [(-1)^2 \cdot 3K] + [0^2 \cdot 7K] + [1^2 \cdot 0.2] + [2^2 \cdot 0.1] \]

Substituting \( K = 0.05 \):

\[ C = [9 \cdot 0.1] + [4 \cdot 0.1] + [1 \cdot 0.15] + [0] + [0.2] + [4 \cdot 0.1] \]

Calculating each term:

\[ C = 0.9 + 0.4 + 0.15 + 0 + 0.2 + 0.4 \]
\[ C = 0.9 + 0.4 + 0.15 + 0.2 + 0.4 = 2.05 \]

Now, we need to subtract \( \mu^2 \):

\[ \mu^2 = (-0.25)^2 = 0.0625 \]

Thus,

\[ C = 2.05 - 0.0625 = 1.9875 \]

### Step 4: Ascending Order of \( A, B, C \)

We have:

- \( A = K = 0.05 \)
- \( B = \mu = -0.25 \)
- \( C = 1.9875 \)

Now, arranging these values in ascending order:

\[ B < A < C \quad \text{or} \quad -0.25 < 0.05 < 1.9875 \]

### Final Result

The ascending order is:

\[ B, A, C \]