I’m a Northwestern Polytechnic Unversity student from China, When I did my Probability theory and mathematical statistics homeworks. I met an issue which is pretty hard to understand. That is:
Hello AI, there’s a distribution function F(x) of the random variable X, which is a strictly monotonic continuous function, Y=F(X) is obedient to a uniform distribution on [0, 1] . Please give me a real-life example to help me understand it. Thanks AI.
Totally, he gave me two answers. The first answer is:
Hello! That’s a good question. A real-life example is rolling dice. Let’s say we have a six-sided dice, each with the same probability of appearing. Then, we can think of each face of the dice as a random variable X with values ranging from 1 to 6. We can define a distribution function F(x) that maps the value of X to the [0,1] interval. For example, F(1)=0, F(2)=1/6, F(3)=2/6, and so on. Then , Y=F(X) obeys a uniform distribution on [0,1]. Hope this example helps you understand the concept.
To be honest, I am more confused by this answer. But the second answer light me.
Of course. Another example is the birthday issue. Let’s say we have a room with 23 people in it. We can think of each person’s birthday as a random variable X with values ranging from 1 to 365 (assuming 365 days per year). We can define a distribution function F(x) that maps the value of X to the [0,1] interval. For example, F(1)=0, F(2)=1/365, F(3)=2/365, and so on. Then , Y=F(X) obeys a uniform distribution on [0,1]. Hope this example helps you understand the concept.
By comparison, we find that the second answer uses numbers like 365, 23, 1/365, 2/365, while the first answer just uses numbers like 6, 1/6, 2/6. So I guess that the example contained a larger amount of data would help us understand the issue better.