Question

The number of cartoons X, watched by Mrs. Kelly's first grade class on Saturday morning is shown below.

X P(X)
0 0.15
1 0.20
2 0.30
3 0.10
4 0.20
5 0.05
Give the standard deviation for the probability distribution above.

Answer 1

The standard deviation \[\sigma\] is the square root of the variance:
\[\sigma = \sqrt{\mathbb{V}(X)} = \sqrt{\mathbb{E}(X^2) - (\mathbb{E}(X)^2})\]where \[\mathbb{E}(X)\]is the expected value of X. By definition,
\[\mathbb{E}(X) = \sum_{x \in D} x p(x)\]where D is the values X may take (in this case \[D = \{0, 1, 2, 3, 4, 5\})\]and \[p(x) = \mathbb{P}(X = x)\]is the probability that X equals x.
Then,
\begin{eqnarray*}\mathbb{E}(X) &=& 0 \cdot p(0) + 1 \cdot p(1) + 2 \cdot p(2) + 3 \cdot p(3) + 4 \cdot p(4) + 5 \cdot p(5) \\
&=& 2.15.
\end{eqnarray*}Similarly,
\begin{eqnarray*}\mathbb{E}(X^2) &=& 0^2 \cdot p(0) + 1^2 \cdot p(1) + 2^2 \cdot p(2) + 3^2 \cdot p(3) + 4^2 \cdot p(4) + 5^2 \cdot p(5) \\
&=& 6.75,
\end{eqnarray*}so
\[\mathbb{V}(X) = 6.75 - 2.15 = 4.6\]and\[\sigma = \sqrt{4.6} \approx 2.144.\]