Question

A distribution that represents the number of cars parked in a randomly selected residential driveway on any night is given by

xi:	0	1	2	3	4
pi :	0.1	0.2	0.35	0.25	0.15

Which of the following statements is correct?

(a) This is a legitimate probability distribution because each of the pi-values is between $0$and $1$.

(b) This is a legitimate probability distribution because $\sum _{}{x}_i$is exactly $10$.

(c) This is a legitimate probability distribution because each of the pi-values is between $0$and $1$and the $\sum _{}{x}_i$is exactly $10$.

(d) This is not a legitimate probability distribution because $\sum _{}{x}_i$is not exactly $10$.

(e) This is not a legitimate probability distribution because role="math" localid="1650523963322" $\sum _{}{p}_i$ is not exactly $1$.

Short answer

As $\sum _{}{p}_i\ne 1$, so the option(e) is correct i.e. This is not a legitimate probability distribution because $\sum _{}{p}_i$is not exactly $1$.

Step-by-step solution

Given information

We are given that,

Distribution of no. of parked cars in a randomly selected residential driveway on any night is given as shown in table given in the question.

Simplify

We know that a legitimate probability distribution needs to have all probabilities $p_i$between $0$to $1$which is satisfied.

Also, a legitimate probability distribution requires the property that the sum of all probabilities $p_i$is equal to $1$.

As sum of probabilities is,

role="math" localid="1650524660369" $$\begin{gathered} \sum _{}{p}_i=0.1+0.2+0.35+0.25+0.15 \\ =1.05\ne 1 \end{gathered}$$

Due to which above property is not satisfied.

Therefore we note that the probability distribution is NOT legitimate.

Hence the correct option is (e).