7. Simulate random variable with given cumulative distribution function

• The cumulative distribution function of a random variable \(X\) on the interval \([0,2]\) is \[ F_X(x) = \begin{cases} 2x^2/9, &\text{if \(x\in[0,1.5]\),}\\ -x^2+4x-3, &\text{if \(x\in(1.5,2]\).} \end{cases} \] Write a Python function that simulates a random variable with such a cumulative distribution function \(F_X\). Write a program that generates \(1000\) such random numbers and prints two numbers in the first line: their average and the relative frequency of the number \(1.5\).
• In this task you have to make a little calculation: determine the exact values that come out from the simulation, that is print out in the second line the exact value of \(\mathbb{E}(X)\), and \(\mathbb{P}(X=1.5)\). Calculating \(\mathbb{E}(X)\) use module sympy or scipy.
• Help: to calculate the expected value from CDF but without the probability distribution function is possible. Namely, if \(X\ge0\), then \(\mathbb{E}(X)=\int_0^\infty 1-F_X(x)\,\mathrm{d}x\) (see e.g. in stackexchange the second and last answer).
• Help: for solving this problem you have to understand, how to generate a random variable from uniform distribution using the inverse CDF (see Wikipedia). So first calculate the inverse of \(F_X\), and think about what shoud be done at the discontinuities (you have to modify the inverse a littlebit).
• The aim of the task is to understand the function of a random variable, for example to understand the random variable \(F_X(X)\), and to use this to simulate a given random variable. The random variable in this task is neither continuous nor discrete!