Chapter 7, Problem 1CR

 (a) Why does a point have zero probability in a continuous distribution? (b) Why are probabilities areas under curves in a continuous distribution?

To determine

Explain the reason that a point has zero probability in a continuous distribution.

Explanation of Solution

In a continuous distribution a random variable can takes infinite number of values.

Now, within a given interval there is infinite number of observations. Hence, the probability to choose a particular value from a given interval tends to zero.

That is, $\frac{1}{\infty }=0$.

Thus, a point has zero probability in a continuous distribution

To determine

Explain the reason that probabilities are areas under curves in a continuous distribution.

Explanation of Solution

A continuous probability distribution is defined as a probability density function.

The area under the curve is equal to 1. The frequency of occurrence of values between any two points equals the total area under the curve between the two points and the horizontal axis.

In case of discrete distribution, the probability of a mass function is the sum of the area of histograms. Now, as a continuous distribution contains infinite number of values, it can be considered as the sum of an infinite number of such histograms, which while drawing, becomes like a smooth curve.

Hence, the probability is defined as area under the continuous curve.