Step 1a

Finding the mean of the random variable

Explanation: Mean E(X) = \$\sum_{i=1}^n x_i p_i(x)\$

E(X) = (1 × 0.15) + (2 × 0.10) + (3 × 0.10) + (4 × 0.01) + (5 × 0.08) + (6 × 0.01) + (7 × 0.05) + (8 × 0.02) + (9 × 0.28) + (10 × 0.20)

E(X) = 0.15 + 0.2 + 0.3 + 0.04 + 0.4 + 0.06 + 0.35 + 0.16 + 2.52 + 2

Mean of the random variable E(X) = 6.18

Step 1b

Finding the variance of the random variable

Explanation: \$VAR(X) = E(X^2) - (E(X))^2\$

E(X) = 6.18

\$E(X^2) = \sum_{i=1}^n (x_i)^2 p_i(x)\$

\$E(X^2) = (1^2 × 0.15) + (2^2 × 0.10) + (3^2 × 0.10) + (4^2 × 0.01) + (5^2 × 0.08) + (6^2 × 0.01) +\$

\$(7^2 × 0.05) + (8^2 × 0.02) + (9^2 × 0.28) + (10^2 × 0.20)\$

⇒ (1 × 0.15) + (4 × 0.10) + (9 × 0.10) + (16 × 0.01) + (25 × 0.08) + (36 × 0.01) + (49 × 0.05) + (64 × 0.02) + (81 × 0.28) + (100 × 0.20)

\$E(X^2)\$ = 50.38

\$VAR(X) = 50.38 - (6.18)^2\$

VAR(X) = 50.38 - 38.19

Variance of the random variable \$VAR(X) = 12.19\$