What is a continuous distribution?

A continuous distribution describes the probabilities of the possible values of a continuous random variable. A continuous random variable is a random variable with a set of possible values (known as the range or support) that is infinite and uncountable.

Probabilities of continuous random variables (X) are defined as the area under the curve of its distribution. Thus, only ranges of values can have a nonzero probability. The probability that a continuous random variable equals some value is always zero.

Example of the distribution of weights

The continuous normal distribution can describe the distribution of weight of adult males. For example, you can calculate the probability that a man weighs between 160 and 170 pounds.

Distribution of the weight of adult males

The shaded region under the curve in this example represents the range from 160 and 170 pounds, and the area of the shaded region equals the probability of this event. The area is 0.136; therefore, the probability that a man weighs between 160 and 170 pounds is 13.6%. The entire area under the curve equals 1.0.

However, the probability that X is exactly equal to some value is always zero because the area under the curve at a single point, which has no width, is zero. For example, the probability that a man weighs exactly 190 pounds to infinite precision is zero. You could calculate a nonzero probability that a man weighs more than 190 pounds, or less than 190 pounds, or between 189.9 and 190.1 pounds, but the probability that he weighs exactly 190 pounds is zero.

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