# Why is the probability zero for any given value of a normal distribution?

I noticed that in the Normal distribution, the probability $P(x=c)$ equals zero, while for the Poisson distribution, it will not equal zero when $c$ is a non-negative integer.

My question is: Does the probability of any constant in the normal distribution equal zero because it represents the area under any curve? Or it is just only a rule to memorize?

• – Glen_b Jun 3 '13 at 5:57
• Very closely related (slightly different question, essentially the same answer): stats.stackexchange.com/questions/4220. – whuber Jun 3 '13 at 13:44
• Nothing worth knowing is ever only a "rule to memorize". – Matthew Drury Dec 6 '17 at 16:03

Perhaps the following thought-experiment helps you to understand better why the probability $Pr(X=a)$ is zero in a continuous distribution: Imagine that you have a wheel of fortune. Normally, the wheel is partitioned in several discrete sectors, perhaps 20 or so. If all sectors have the same area, you would have a probability of $1/20$ to hit one specific sector (e.g. the main price). The sum of all probabilities is 1, because $20\cdot 1/20 = 1$. More general: If there are $m$ sectors evenly distributed on the wheel, every sectors has a probability of $1/m$ of being hit (uniform probabilities). But what happens if we decided to partition the wheel into a million sectors. Now the probability of hitting one specific sectors (the main prize), is extremely small: $1/10^{6}$. Further, note that the pointer can theoretically stop at an infinite number of positions of the wheel. If we wanted to make a separate prize for each possible stopping point, we would have to partition the wheel in an infinite number of "sectors" of equal area (but each of those would have an area of 0). But what probability should we assign to each of these "sectors"? It must be zero because if the probabilities for each "sectors" would be positive and equal, the sum of infinitely many equal positive numbers diverges, which creates a contradiction (the total probability must be 1). That's why we only can assign a probability to an interval, to a real area on the wheel.

More technical: In a continuous distribution (e.g. continuous uniform, normal, and others), the probability is calculated by integration, as an area under the probability density function $f(x)$ (with $a\leq b$): $$P(a\leq X \leq b) = \int_{a}^{b} f(x) dx$$ But the area of an interval of length 0 is 0.

See this document for the analogy of the wheel of fortune.

The Poisson distribution on the other hand is a discrete probability distribution. A random Poisson variable can only take discrete values (i.e. the number of children for one family cannot be 1.25). The probability that a family has exactly 1 child is certainly not zero but is positive. The sum of all probabilities for all values must be 1. Other famous discrete distributions are: Binomial, negative binomial, geometric, hypergeometric and many others.

• This argument fails at a crucial point: it is not always the case that "the sum of an infinity number of positive numbers is infinite." The sequence of Poisson probabilities is a counterexample! You can fix this up by a suitable qualification, such as pointing out that the sum of infinitely many positive numbers, no matter how small they might be, diverges. – whuber Dec 5 '17 at 22:11
• @whuber I think that's what I meant back when I wrote the answer but failed to formulate it properly. Thanks for the heads up. I hope it's correct now. – COOLSerdash Dec 6 '17 at 10:57
• Thank you. However, you still adduce a false statement: "the sum of infinitely many positive numbers diverges." The sum of infinitely many positive Poisson probabilities is $1$, as a counterexample. – whuber Dec 6 '17 at 14:32
• @whuber Now I'm confused. That's exactly the formulation you suggested I add in your first comment: "[...] such as pointing out that the sum of infinitely many positive numbers, no matter how small they might be, diverges" – COOLSerdash Dec 6 '17 at 14:42
• @whuber Right, now it's fully clear. I added the qualification to my answer. Thanks again for pointing it out. – COOLSerdash Dec 6 '17 at 14:45

"Probabilities of continuous random variables (X) are defined as the area under the curve of its PDF. Thus, only ranges of values can have a nonzero probability. The probability that a continuous random variable equals some value is always zero." reference page: http://support.minitab.com/en-us/minitab-express/1/help-and-how-to/basic-statistics/probability-distributions/supporting-topics/basics/continuous-and-discrete-probability-distributions/