Exercise 5.1.5

• We see that the number ending with $0$ will still end with $0$ after square.

• The numbers ending with $1,9$ will end with $1$ after square.

• The numbers ending with $2,8$ will end with $4$ after square.

• The numbers ending with $5$ will end with $5$ after square.

• The numbers ending with $4,6$ will end with $6$ after square.

• The numbers ending with $3,7$ will end with $9$ after square.

So we have $p_0=\frac{100}{1000}=\frac{1}{10},p_1=p_4=p_6=p_9=\frac{200}{1000}=\frac{2}{10},p_5=\frac{100}{1000}=\frac{1}{10}$

Then the mean $m=(0+5)$\times \frac{1}{10}+(1+4+6+9)\times \frac{2}{10}=4.5\$Thevariance$ \sigma 2=\; (0\; +\; 52)\; \times \frac{1}{10}+\; (12+\; 42+\; 62+\; 92)\; \times \frac{2}{10}=\; 29.3$$