问题及解答

The pmf for $X=$ the number of major defects on a randomly selected appliance of a certain type is

Compute the following:

• (a) $E(X)$
• (b) $V(X)$ directly from the definition
• (c) The standard deviation of $X$
• (d) $V(X)$ using the shortcut formula
   

(a) 

\[
E(X)=\sum_{x=0}^{4}x\cdot p(x)=0\times 0.08+1\times 0.15+2\times 0.45+3\times 0.27+4\times 0.05=2.06
\]

(b) Calculate variance by the definition,

\[
\begin{split}
V(X)&=E[(X-\mu)^2]=\sum_{x=0}^{4}(x-2.06)^2\cdot p(x)\\
&=(0-2.06)^2\cdot 0.08+(1-2.06)^2\cdot 0.15+(2-2.06)^2\cdot 0.45+(3-2.06)^2\cdot 0.27+(4-2.06)^2\cdot 0.05\\
&=0.9364
\end{split}
\]

(c)

The standard deviation (SD 标准差/標準差)

\[
\sigma_X=\sqrt{\sigma_X^2}=\sqrt{V(X)}=\sqrt{0.9364}\approx 0.9677
\]

(d) Using the formula $V(X)=E(X^2)-(E(X))^2$

\[
\begin{split}
V(X)&=E(X^2)-(E(X))^2\\
&=\sum_{x=0}^{4}x^2 p(x)-2.06^2\\
&=0^2\times 0.08+1^2\times 0.15+2^2\times 0.45+3^2\times 0.27+4^2\times 0.05-2.06^2\\
&=5.18-2.06^2\\
&=0.9364
\end{split}
\]