Scale parameter in probability distributions

A parameter s is called a scale parameter of a probability distribution when it follows the following property:

F(x|s,θ) = F(x/s | 1,θ)

Here, F is the cumulative distribution function (cdf) and θ represents other parameters in the distribution.

When the probability distribution function (pdf) is defined for all values, then s must follow the following property:

f_s(x) = f(x/s)/s

Here, f is pdf of the standard distribution, and  f_s is pdf of the scaled distributions.

Intuitively,  higher scale parameter means higher spread of the distribution. Standard normal distribution with different scales are shown in the following plot.

And, here is matlab code to generate the above plot.

scales = [5,2,1,0.5,0.3];
x = -10:.001:10;
for s = scales
    y = normpdf(x./s) ./ s ;
    plot(x,y);
    hold on
end
legend('scale=5', 'scale=2', 'scale=1', 'scale=0.5', 'scale=0.3')
title('Standard normal distribution with different scale parameters')