Complete parts (a) and (b) below.
The number of dogs per household in a small town
Dogs
Probability
0
1
3
0.220
2
5
0.027
$\begin{array}{cc}4 & 5 \\ 0.014 & 0.008\end{array}$
(a) Find the mean, variance, and standard deviation of the probability distribution.
Find the mean of the probability distribution.

Step 1 ：The mean of a probability distribution is calculated by multiplying each outcome by its probability and then summing these products. In this case, the mean (μ) is calculated as follows: μ = 0*0.220 + 1*0.027 + 2*0.014 + 3*0.008 + 4*0.005 = 0.220 + 0.027 + 0.028 + 0.024 + 0.020 = 0.319

Step 2 ：The variance of a probability distribution is calculated by subtracting the mean from each outcome, squaring the result, multiplying by the probability of the outcome, and then summing these products. In this case, the variance (σ²) is calculated as follows: σ² = (0-0.319)²*0.220 + (1-0.319)²*0.027 + (2-0.319)²*0.014 + (3-0.319)²*0.008 + (4-0.319)²*0.005 = 0.022 + 0.005 + 0.004 + 0.005 + 0.006 = 0.042

Step 3 ：The standard deviation of a probability distribution is the square root of the variance. In this case, the standard deviation (σ) is calculated as follows: σ = √0.042 = 0.205

Step 4 ：So, the mean of the probability distribution is 0.319, the variance is 0.042, and the standard deviation is 0.205.