Question

Random variable x has a normal distribution with parameters μ=3.3 and σ=1.3. What is the probability that a randomly selected x is equal to a=0?

Pn(x=a)=?

How would I calculate this is there a rule/formula

Answer 1

Rule : Probability of any constant in the continuous distribution equal zero

Reasoning : 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.

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 ( ) :

But the area of an interval of length 0 is 0

So, in case of probability at any point , the length of the interval is zero. This means that Area of that interval is Zero.

Since, Area under the curve represents the probability, we can say that Probability of any constant in the continuous distribution equal zero.

P(X=a) = 0