This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — here's how to determine the mean, variance, and standard deviation for this discrete probability distribution: Step 1: Calculate the mean (E(X)). The mean (or expected value) is calculated using the formula E(X) = x · p(x). E(X) = (3 × 0.2) + (8 × 0.7) + (10 × 0.1) E(X) = 0.6 + 5.6 + 1.0 E(X) = 7.2 The mean is 7.2. Step 2: Calculate E(X^2). This is needed for the variance calculation. The formula is E(X^2) = x^2 · p(x). E(X^2) = (3^2 × 0.2) + (8^2 × 0.7) + (10^2 × 0.1) E(X^2) = (9 × 0.2) + (64 × 0.7) + (100 × 0.1) E(X^2) = 1.8 + 44.8 + 10.0 E(X^2) = 56.6 Step 3: Calculate the variance (Var(X)). The variance is calculated using the formula Var(X) = E(X^2) - [E(X)]^2. Var(X) = 56.6 - (7.2)^2 Var(X) = 56.6 - 51.84 Var(X) = 4.76 The variance is 4.76. Step 4: Calculate the standard deviation (_X). The standard deviation is the square root of the variance. _X = sqrt(Var(X)) _X = sqrt(4.76) _X ≈ 2.1817 The standard deviation is 2.18 (rounded to two decimal places). One more to go! Send it over.