Question

5.1 basics of probability distributions. compute the mean and standard deviation of a discrete random variable.
the following table shows the number of people that have either 0, 1, 2 or 3 cars registered in their name, from a sample of 270 people.
a) complete the table.

x (the number of cars)	number of people
1	80
2	78
3	i

b) find the mean number of cars owned for this sample. give your answer to at least 2 decimal places.
c) find the standard deviation of cars owned for this sample. give your answer to at least 2 decimal places.
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Explanation:

Step1: Calculate the number of people with 3 cars

The total number of people in the sample is 270. Let the number of people with 3 cars be $n$. Then $n=270-(62 + 80+78)=270 - 220 = 50$.

Step2: Calculate the mean $\mu$

The formula for the mean of a discrete - random variable is $\mu=\sum_{i}x_ip_i$. Here, $x$ is the number of cars and $p$ is the probability. The probabilities $p_i$ are calculated as $p_i=\frac{\text{Number of People}}{\text{Total Number of People}}$.
\[

$$\begin{align*} \mu&=\frac{0\times62 + 1\times80+2\times78 + 3\times50}{270}\\ &=\frac{0 + 80+156+150}{270}\\ &=\frac{386}{270}\approx1.43 \end{align*}$$

\]

Step3: Calculate the variance $\sigma^{2}$

The formula for the variance of a discrete - random variable is $\sigma^{2}=\sum_{i}(x_i-\mu)^{2}p_i$.
\[

$$\begin{align*} \sigma^{2}&=\frac{(0 - 1.43)^{2}\times62+(1 - 1.43)^{2}\times80+(2 - 1.43)^{2}\times78+(3 - 1.43)^{2}\times50}{270}\\ &=\frac{( - 1.43)^{2}\times62+( - 0.43)^{2}\times80+(0.57)^{2}\times78+(1.57)^{2}\times50}{270}\\ &=\frac{2.0449\times62 + 0.1849\times80+0.3249\times78+2.4649\times50}{270}\\ &=\frac{126.7838+14.792+25.3422 + 123.245}{270}\\ &=\frac{290.163}{270}\approx1.07 \end{align*}$$

\]

Step4: Calculate the standard deviation $\sigma$

The standard deviation is the square - root of the variance. So $\sigma=\sqrt{\sigma^{2}}=\sqrt{1.07}\approx1.03$

Answer:

a) 50
b) 1.43
c) 1.03