# Central limit rule-of-thumb for Binomial approximation

Hi MIT proba OpenCourseWare reading-questions-6b first: I think maybe there is a mistake in the answer given for the problem 4: Exact: 0.0227, rule-of-thumb: 0.025 ? second : can someone tell me what is wrong in my raisonning , and how calculate the rule-of-thumb ? third: why I get an error if I uncomment the last line below ?

```
forget()
#############################
import matplotlib.pyplot as plt
def makeFig(pdfTable,cdfTable,textFig):
# Make a figure
fig = plt.figure()
# Make room for legend at bottom
fig.subplots_adjust(bottom=0.2)
# The axes for your lists 1-2
ax1 = fig.add_subplot(211)
# Plot lines 1-3
line1 = ax1.plot([0] + pdfTable,'bo-',label='pdf')
line2 = ax1.plot([0] + cdfTable,'go-',label='cdf')
# To get lines 1-2 on the same legend, we need to
# gather all the lines together before calling legend
lines = line1+line2
labels = [l.get_label() for l in lines]
# giving loc a tuple in axes-coords. ncol=2 for 2 columns
ax1.legend(lines, labels, loc=(0,-0.4), ncol=2)
ax1.set_xlabel('--------'+ textFig)
# Display the figure
plt.show()
return
######################
n=64
kMax=40
p=0.5
kTable=[i for i in range(0,kMax+1)]
pdfTable=[]
cdfTable=[]
#A fair coin is tossed 64 times. Using Binomial Distribution to estimate the probability of getting more that 40 heads.
for k in kTable:
pdfTable.append((binomial(n,k)*p^k *(1-p)^(n-k)))
cdfTable.append(sum(pdfTable))
#pdfTableR = [ round(elem, 5) for elem in pdfTable ]
#show(pdfTableR)
cdfTableR = [ round(elem, 5) for elem in cdfTable ]
show(cdfTableR)
makeFig(pdfTable,cdfTable,' P more than 40 Heads On 64 toss')
pSupKmax=1-cdfTable[-1]
show('P(X > 40 Heads) On 64 toss ')
show('Using Binomial distribution : ',pSupKmax)
#A fair coin is tossed 64 times. Use the central limit theorem to estimate the probability of getting more that 40 heads.
################################
# for one toss
mu_1=p # Expectation for one Bernouilli
sigma_1=sqrt(p*(1-p)) # Standard deviation for one Bernouilli
# for 64 toss, sum of 64 bernoulli variables
mu_64=n*mu_1 # expectation= sum of the 64 Bernoulli Expectation
sigma_64=sqrt(n)*sigma_1 # Standard Deviation= sum of the 64 Bernoulli standard Deviation
# rule-of-thumb
show('Using 1-erf :', 1-erf( ((kMax-mu_64)/sigma_64) ) )
#############################################################
#using Central Limit Theorem using normal law integration
#############################################################
#from sage.symbolic.integration.integral import indefinite_integral
from sage.symbolic.integration.integral import definite_integral
x = var('x')
assume(x, 'real')
normalPDF = function('normalPDF')(x)
normalCDF = function('normalCDF')(x)
# normal PDF(x,mu_0,sigma_0)
normalPDF=1/(2 *sqrt(pi*sigma_64^2))*e^(-(((x-mu_64)/sigma_64 )^2)/2)
normalCDFnum=definite_integral(normalPDF,x,-infinity,kMax)
show('Using normal law integration 1-normalCDFnum :',(1-normalCDFnum ))
# error if line below uncommented
#show('Using normal law integration 1-normalCDFnum :',(1-normalCDFnum ).n())
```

Ooops the third question is closed !, I made a mistake using sigma_0 :-( it must be :

normalPDF=1/(2

sqrt(pisigma_64^2)) * e^(-(((x-mu_64)/sigma_64 )^2)/2)not :

normalPDF=1/(2

sqrt(pisigma_0^2)) * e^(-(((x-mu_64)/sigma_64 )^2)/2)i have now edited the code. sorry

it remains the first and second question, but it is pure mathematical knowledge now. and maybe it is not the place to post it. Maybe I should delete this post and ask on math forum now ?