Question

Compute the mean and variance of the following discrete probability distribution. (Round your answers to 2 decimal places.) X P(X) 0 .2 1 .4 2 .3 3 .1

Answer 1

The given distribution is as follows:

\(\begin{array}{ll}X & P(X) \\ 0 & 0.2 \\ 1 & 0.4 \\ 2 & 0.3 \\ 3 & 0.1\end{array}\)

Calculate mean as follows:

$$ \begin{aligned} \bar{X} &=\sum_{X} X \times P(X) \\ &=(0 \times 0.2)+(1 \times 0.4)+(2 \times 0.3)+(3 \times 0.1) \\ &=0+0.4+0.6+0.3 \\ &=1.3 \end{aligned} $$

Calculate variance as follows:

$$ \begin{aligned} V(X) &=\left[\sum_{X} X^{2} P(X)\right]+(\bar{X})^{2} \\ &=\left[\left(0^{2} \times 0.2\right)+\left(1^{2} \times 0.4\right)+\left(2^{2} \times 0.3\right)+\left(3^{2} \times 0.1\right)\right]-(1.3)^{2} \\ &=0+0.4+1.2+0.9-1.69 \\ &=0.81 \end{aligned} $$