If we plot a continuous probability distribution f(x),the total probability under the curve is
A) -1
B) 0
C) 1
D) 100
Suppose that the number of customers arriving each hour at the only checkout counter at a local convenience store is approximately Poisson distributed with an expected arrival rate of 30 customers per hour.Let X represent the number of customers arriving per hour.The probabilities associated with X are shown below.
P(X

Question

If we plot a continuous probability distribution f(x),the total probability under the curve is
A) -1
B) 0
C) 1
D) 100
Suppose that the number of customers arriving each hour at the only checkout counter at a local convenience store is approximately Poisson distributed with an expected arrival rate of 30 customers per hour.Let X represent the number of customers arriving per hour.The probabilities associated with X are shown below.
P(X < 5)= 0.0000,P(X < 10)= 0.0000,P(X < 15)= 0.0009,
P(X < 20)= 0.0219,P(X < 25)= 0.1572,P(X < 30)= 0.4757
P(X = 30)= 0.0726,P(X = 31)= 0.0703,P(X = 32)= 0.0659,
P(X = 33)= 0.0599,P(X = 34)= 0.0529,P(X = 35)= 0.0453

Quizplus · Accepted Answer

The total probability under the curve of a continuous probability distribution is always equal to 1, as it represents the entire range of possible outcomes, which must account for all probabilities. Therefore, the correct answer is C.

The probabilities associated with X in the provided distribution represent the likelihood of a certain number of customers arriving per hour at the convenience store. Based on this distribution, the following can be inferred:

- The probability of fewer than 5 customers arriving in an hour is essentially 0, as P(X < 5) = 0.0000.
- Similarly, the probabilities of less than 10 and 15 customers arriving are also negligible.
- The probability of 20 or more customers arriving is becoming increasingly more likely, as P(X < 20) = 0.0219 and P(X < 25)= 0.1572.
- The most probable number of customers arriving per hour is 30, as P(X=30) = 0.4757, and the probabilities of smaller or larger values are decreasing.
- The probabilities of X being equal to or larger than 31 are relatively small, so it is unlikely for more than 30 customers to arrive in an hour.

Based on these observations, the best choice(s) would be to staff the checkout counter to handle a rate of 30 customers per hour, as this is the most probable rate of arrival. If the store wants to be prepared for higher traffic, they could staff the counter to handle 35 customers per hour, as the probability of 35 customers arriving in an hour is still reasonably high at 0.0453.

If We Plot a Continuous Probability Distribution F(x),the Total Probability