Question

Kids and toys Refer to Exercise 4. Calculate and interpret the standard deviation of the random variable $X$. Show your work.

Short answer

The number of toys played with is on average $1.3106$ toys from the mean.

Step-by-step solution

Given Information

Given in the question that, the probability distribution of the number $X$of toys played with by a randomly selected subject is as follows:

Explanation

The expected value is computed by adding each possibility by its probability:

$$\begin{gathered} E\left(X\right)=\sum xP\left(x\right) \\ =0\times 0.03+1\times 0.16+2\times 0.30+3\times 0.23+4\times 0.17+5\times 0.11 \\ =2.68 \end{gathered}$$

The expected value of the squared variation from the mean is the variance:

$$\begin{gathered} {\sigma }^2=\sum (x-\mu {)}^{2}P(x) \\ =(0-2.68{)}^2\times 0.03+(1-2.68{)}^2\times 0.16+(2-2.68{)}^2\times 0.30+(3-2.68{)}^2\times 0.23 \end{gathered}$$

The standard deviation is the square root of the variance:

$$\begin{gathered} \sigma =\sqrt{{\sigma }^2} \\ =\sqrt{1.7176} \\ \approx 1.3106 \end{gathered}$$