Goal: Determine the probability P(X=0) when a random variable X takes on the values -1, 0, and 1 with a mean of 0.6.

Given:
- Random variable X takes on the values -1, 0, and 1.
- The mean of X is 0.6.
- P(X=0) = 0.2.

Solution:

Step 1: Understanding the Mean of a Random Variable
The mean of a random variable is a measure of its central tendency and represents the average value of the variable. It is calculated by summing the products of each possible value of the variable and its corresponding probability.

Step 2: Calculating the Mean
Given that the mean of X is 0.6, we can set up the equation:
Mean = (-1 * P(X=-1)) + (0 * P(X=0)) + (1 * P(X=1))

Since P(X=0) = 0.2, we can rewrite the equation as:
0.6 = (-1 * P(X=-1)) + (0 * 0.2) + (1 * P(X=1))

Simplifying the equation:
0.6 = -P(X=-1) + P(X=1)

Step 3: Using the Given Values
Since the possible values of X are -1, 0, and 1, and P(X=0) = 0.2, we can substitute these values into the equation:
0.6 = (-1 * P(X=-1)) + (0 * 0.2) + (1 * P(X=1))

Simplifying the equation further:
0.6 = -P(X=-1) + P(X=1)

Step 4: Solving for P(X=-1) and P(X=1)
To solve for P(X=-1) and P(X=1), we can isolate each probability term:
0.6 + P(X=-1) = P(X=1)

Since the sum of probabilities must equal 1, we have:
P(X=-1) + P(X=0) + P(X=1) = 1

Substituting the given values:
P(X=-1) + 0.2 + P(X=1) = 1

Replacing P(X=1) with 0.6 + P(X=-1):
P(X=-1) + 0.2 + (0.6 + P(X=-1)) = 1

Simplifying the equation:
1.8 + 2P(X=-1) = 1

Rearranging the equation:
2P(X=-1) = 1 - 1.8
2P(X=-1) = -0.8

Finally, solving for P(X=-1):
P(X=-1) = -0.8 / 2
P(X=-1) = -0.4

Step 5: Calculating P(X=0)
We can now calculate P(X=0) using the sum of probabilities equation:
P(X=-1) + P(X=0) + P(X=1) = 1

Substituting the values we have found:
-0