The sum ~ Normal(mean_sum = n*μ, std_sum = sqrt(n)*σ)
print(f"{p1:.3f}") print(f"{p2:.3f}") print(f"{p3:.3f}") 0.401 0.159 0.440 Variation: Central Limit Theorem Problem Sometimes Problem 6 asks: A large number of i.i.d. random variables each with mean μ and variance σ². Find the probability that the sum of n variables exceeds a value S. Solution using CLT : probability and statistics 6 hackerrank solution
[ \Phi(z) = \frac{1}{2} \left[ 1 + \text{erf}\left(\frac{z}{\sqrt{2}}\right) \right] ] HackerRank allows math.erf() and math.sqrt() . Here's a clean solution: The sum ~ Normal(mean_sum = n*μ, std_sum =
If you're working through HackerRank's 10 Days of Statistics or their Probability and Statistics challenges, Problem 6 usually introduces the Normal Distribution (Gaussian Distribution) and sometimes the Central Limit Theorem (CLT) . Solution using CLT : [ \Phi(z) = \frac{1}{2}
[ P(X \leq x) = \Phi\left(\frac{x - \mu}{\sigma}\right) ]
import math def normal_cdf(x, mean, std): """Cumulative probability P(X <= x) for N(mean, std)""" z = (x - mean) / std return 0.5 * (1 + math.erf(z / math.sqrt(2))) mean, std = 20, 2 x1, x2 = 19.5, 22 1. P(X <= x1) p1 = normal_cdf(x1, mean, std) 2. P(X >= x2) p2 = 1 - normal_cdf(x2, mean, std) 3. P(x1 <= X <= x2) p3 = normal_cdf(x2, mean, std) - normal_cdf(x1, mean, std)
where (\Phi(z)) is the CDF of the standard normal distribution. We can compute (\Phi(z)) using the :