Question

complete parts (a) and (b) below.
the number of dogs per household in a small town
dogs 0 1 2 3 4 5
probability 0.687 0.196 0.075 0.021 0.013 0.008

(a) find the mean, variance, and standard deviation of the probability distribution.
find the mean of the probability distribution.
μ = 0.5 (round to one decimal place as needed.)
find the variance of the probability distribution.
σ² = (round to one decimal place as needed.)

Explanation:

Step1: Recall variance formula

The formula for the variance $\sigma^{2}$ of a discrete - probability distribution is $\sigma^{2}=\sum(x - \mu)^{2}P(x)$, where $x$ is the value of the random variable, $\mu$ is the mean, and $P(x)$ is the probability of $x$. We know $\mu = 0.5$.

Step2: Calculate $(x-\mu)^{2}P(x)$ for each $x$

For $x = 0$: $(0 - 0.5)^{2}\times0.687=( - 0.5)^{2}\times0.687 = 0.25\times0.687=0.17175$
For $x = 1$: $(1 - 0.5)^{2}\times0.196=(0.5)^{2}\times0.196 = 0.25\times0.196 = 0.049$
For $x = 2$: $(2 - 0.5)^{2}\times0.075=(1.5)^{2}\times0.075=2.25\times0.075 = 0.16875$
For $x = 3$: $(3 - 0.5)^{2}\times0.021=(2.5)^{2}\times0.021 = 6.25\times0.021=0.13125$
For $x = 4$: $(4 - 0.5)^{2}\times0.013=(3.5)^{2}\times0.013 = 12.25\times0.013 = 0.15925$
For $x = 5$: $(5 - 0.5)^{2}\times0.008=(4.5)^{2}\times0.008 = 20.25\times0.008=0.162$

Step3: Sum up the values

$\sigma^{2}=0.17175 + 0.049+0.16875 + 0.13125+0.15925 + 0.162$
$\sigma^{2}=0.842$
Rounding to one decimal place, $\sigma^{2}=0.8$

Answer:

$0.8$