import math

lambd = 1.2

# 1.p(X = 2 ) 
k = 2
q_1 = (math.e ** (-lambd) * lambd ** k) / math.factorial(k)
print(round(q_1,2))

# 2.p (x < 3)
#  sum(x: 1-> 2)(possion(x, 1.2))
q_2 = sum([(math.e ** (-lambd) * lambd ** k) / math.factorial(k) for k in [0,1,2]])

print(round(q_2,2))

# 3p(x = 5) for /10 pages
lamdbd_ = lambd * 10
k = 5
q_3 = (math.e ** (-lamdbd_) * lamdbd_ ** k) / math.factorial(k)
print(round(q_3,2))


lamdbd_ = lambd * 40
q_4 = sum([(math.e ** (-lamdbd_) * lamdbd_ ** k) / math.factorial(k) for k in [0,1,2,3]])
print(1 -round(q_4,2))

import math
def fact(n):
    return 1 if n == 0 else n*fact(n-1)
def poi(k,lam):
    return ((lam**k)*(math.exp(-lam)))/(fact(k))

#1
lam = 1.2
print(round(poi(2,1.2),3))

#2
print(round(poi(0,1.2) + poi(1,1.2) + poi(2,1.2),3))
#or
k = 2; lam = 1.2
round(sum([poi(x,lam) for x in range(0,k+1)]),3)

#3
lam = 12; k = 5 
print(round(poi(k,lam),3))

#4
k = 40; lam = 1.2
print(round(sum([poi(x,lam) for x in range(0,k+1)]),3))