Use python

As we watch trees lose their leaves, let's try to model this problem.

a$)$ Let's firstly assume that a tree loses all its leaves over a four week period, with a given leaf lost at any time during this period. Assuming a tree has $10000$ leaves, what is the probability that a leaf is lost on day $20.$ Use a numerical simulation, and numpy, to answer this question.

b$)$ Create an array of times between leaf drops $($the numpy command 'diff' may be helpful for this$).$ What is the distribution of time between leaf drops?

Show the distribution on an appropriate scale to justify your conclusion about the distribution type.

c$)$ Let's assume instead that the leaves fall according to a normal distribution around a peak day of leaf loss. Set up your normal distribution so that for the $10000$ leaves, $1$ leaf is expected to fall on day $1,$ and one leaf on day $29,$ and the peak of leaf loss occurs on day $15.$ How many leaves are expected to fall

on day $15?$

d$)$ In a patch of ground beneath deciduous trees, the depth of the leaf litter in $m$ can be modelled by the following integral:

$D(t)={\displaystyle \underset{0}{\overset{t}{}}}0\mathrm{.}2(11-)e^{-\frac{(-10{)}^2}{10}}d$

If $=0\mathrm{.}1$ plot the depth of leaf litter as a function of time by numerically solving the integral in the range $tin[0,20].$

e$)$ Identify when the depth of leaves is $20cm.$