Question

The probability distribution of a random variable X is given. x 1 2 3 4 P(X = x) 0.4 0.1 0.3 0.2 Compute the mean, variance, and standard deviation of X. (Round your answers to two decimal places.) mean variance standard deviation

Answer 1

Mean:

`E(X) = \displaystyle \sum_{x\in\{1,2,3,4\}}x\,P(X=x) = 1*0.4 + 2*0.1 + 3*0.3 + 4*0.2 = \boxed{2.3}`

Variance:

`\displaystyle V(X) = E\left((X-E(X))^2\right) = E(X^2) - E(X)^2 \\\\ E(X^2) = \sum_{x\in\{1,2,3,4\}}x^2\,P(X=x) = 1^2*0.4 + 2^2*0.1 + 3^2*0.3 + 4^2*0.2 = 6.7 \\\\ \implies V(X) = 6.7 - 2.3^2 = \boxed{1.41}`

Standard deviation:

`\sigma_X = √(V(X)) = √(1.41) \approx \boxed{1.19}`