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  1. Prepare
  2. Artificial Intelligence
  3. Probability & Statistics - Foundations
  4. The Central Limit Theorem #5
  5. Discussions

The Central Limit Theorem #5

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8 Discussions

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  • faezeh_jahanmiri
     + 0 comments

    74+11(47-X)<20 X>51.91 Then apply P(X>51.91) on a normal distribution of N(50,10/sqrt(11)), and you'll get 0.2632

  • vai_suliafu
     + 0 comments

    If we start with 74,000 gallons and need to have at least 20,000 gallons after 11 weeks, then we cannot afford to lose more than (74,000 - 20,000)/11 each week.

    We also know that if we sell on average 50,000 gallons each week and add a constant 47,000 gallons each week, then on average we can expect to lose 3,000 gallons each week.

    Now knowing that on average we will lose 3,000 gallons each week, what is the probability we will lose more than (74,000-20,000)/11 each week?

  • deankiss
     + 0 comments
    import math
def cdf(x, mu, sigma):
    return 0.5 + math.erf((x-mu)/(sigma*2**0.5))*0.5

mu = 50000
sigma = 10000

weeks=11
start = 74000
weekly = 47000
end = start + weekly*weeks
lower_bound = end - 20000

sum_mu = weeks*mu
sum_sigma = weeks**0.5*sigma

print('{:0.4f}'.format(1-cdf(lower_bound, sum_mu, sum_sigma)))

  • kskk02
     + 0 comments

    The distribution of the remaining supply is really not normal anymore and is only positive since you clearly cant have a negative balance. My approach is to explicitly simulate the weekly remainder but having a check for less than 0 remainder, in which case, I force it to take the weekly delivered amount as the left over amount for that week.

    remainder = 0
num_exp=100000
delivery = 47
res = []
for exp in range(1,num_exp):
    start = 74
    remainder = start
    for week in range(1,12):
        remainder = remainder - np.random.normal(50, 10)
        if (remainder <=0):
            remainder = delivery
        else:
            remainder += delivery
    res.append(remainder)

  • Fandekasp
     + 1 comment

    I'm stucked at the following challenge #6 because this one's implementation is wrong, as anything > 0.1999 will pass.

    Can someone look at my implementation and tell me where I'm failing?

    from math import sqrt
from scipy.stats import norm

init_supply = 74000
mu = 50000.
sigma = 10000
weeks = 11
delivery = 47000
sample_mu = 20000.

mu_11w = init_supply + (delivery-mu)*weeks
sigma_11w = sigma * weeks

diff_mu = sample_mu - mu_11w
z = diff_mu / sigma_11w
print(format(norm.cdf(z), '.4f'))

    Below the calculations to support the choice of my equations (init_supply → supply, delivery → X, weeks → w)

    mu_11w = supply + (X-mu)*w

sigma_11w = (supply + (X-mu+sigma)*w) - (supply + (X-mu)*w)
          = supply + X*w - mu*w + sigma*w - supply - X*w + mu*w
          = sigma*w

diff_mu =  sample_mu - mu_11w

z = diff_mu * sqrt(N) / sigma_11w
N=1 ⇒ z = diff_mu / sigma_11w