A random variable X follows a probability distribution with the probabilities P(X = 1) = 0.2, P(X = 2) = 0.3, P(X = 3) = 0.5. Find the variance of the distribution.

Step 1 ：The variance of a probability distribution is given by \[Var(X) = E(X^2) - [E(X)]^2\].

Step 2 ：Step1: We first find the expected value E(X), which is given by \[E(X) = \sum xP(X = x)\]. So, \[E(X) = 1*0.2 + 2*0.3 + 3*0.5 = 2.3\]

Step 3 ：Step2: We then find the expected value of the square of the random variable E(X^2), which is given by \[E(X^2) = \sum x^2P(X = x)\]. So, \[E(X^2) = 1^2*0.2 + 2^2*0.3 + 3^2*0.5 = 7.1\]

Step 4 ：Step3: We substitute these values into the formula for variance to find the variance. \[Var(X) = E(X^2) - [E(X)]^2 = 7.1 - (2.3)^2 = 2.51\]