19.3 Real-valued distributions

So far, we have talked about discrete random variables, that is, random variables with countably many values. However, not all experiments/observations/measurements are like this. For instance, the height of a person is a random variable that can assume a continuum of values.

To give a tractable example, let’s pick a number X from [0,1], with each one having an “equal chance.” In this context, equal chance means that

Can we describe X with a single real function? As in the discrete case, we can try

but this wouldn’t work. Why?

Because for each x ∈X, we have P(X = x) = 0. That is, picking a particular number x has zero probability. Instead, we can try F_X(x) = P(X ≤x), which is

We can plot this for visualization:

from scipy.stats import uniform 
X = np.linspace(-0.5, 1.5, 100) 
y = uniform.cdf(X) 
 
 
with plt.style.context(’seaborn-v0_8’): 
    plt.figure(figsize=(10, 5)) 
    plt.title...