How can I save a figure to PDF with a specific page size and padding?

I have generated a matplotlib figure that I want to save to a PDF. So far, this is straightforward.

import matplotlib.pyplot as plt
x = [1, 2, 3, 4]
y = [3, 5, 4, 7]
plt.scatter(x, y)
plt.savefig(
    "example.pdf",
    bbox_inches = "tight"
)
plt.close()

However, I would like the figure to appear in the middle of a standard page and have some white space around the figure, rather than fill up the entire page. How can I tweak the above to achieve this?

Another option is to use constrained layout and set a rectangle that you want the all the plot elements to be contained in.

import matplotlib.pyplot as plt

x = [1, 2, 3, 4]
y = [3, 5, 4, 7]

fig, ax = plt.subplots(figsize=(8.3, 11.7), layout='constrained')
fig.get_layout_engine().set(rect=(0.15, 0.3, 0.7, 0.4))  # left, bottom, width, height in fractions of the figure

ax.scatter(x, y)

fig.savefig('example.pdf')

plt.close()