Your question is Validate a Probability Distribution. Take a moment with it on the right.
Talk me through your thinking if you like. When you're confident, submit your answer and I'll grade it like a real screen (7/10 or better passes).
DRW's market-making desk rescales realized slippage on a strategy (in basis points) onto x in [-1, 1] and someone proposes the following candidate density, along with a Monte Carlo sanity check a quant ran before bringing it to you.
Candidate density (claimed to already be normalized):
f(x) = (3/2) * (1 - x^2), for x in [-1, 1]
f(x) = 0, otherwise
Monte Carlo check (n = 2000 uniform draws over [-1, 1], estimating the
integral of f over its support):
estimate of the integral of f(x) over [-1, 1]: 1.98
95% confidence interval: [1.95, 2.01]
Decide whether f is a valid probability density function. Use both an exact analytic check (non-negativity and normalization) and the Monte Carlo confidence interval above, and explain what each one tells you that the other doesn't. Then explain precisely how you would repair f so it's valid.