Question

(c) A random sample of size 30 is taken from Ρo (4) (i.e. Poisson distribution). Find
i. Ρ ( Χ < 4.5)
ii. Ρ ( Χ > 3.8)
iii. Ρ (3.8 < Χ < 4.5)

Answer 1

$$f(x) = \lambda ^x exp(x) / x!$$

but

$$\lambda = 4$$

i. P(x<4.5)

Solution

P(x<4.5)= P(x=0) + P(x=1) + P(x=2) + P(x=3) + P(x=4)

$$P(x=0)= 4^0 exp(-4) / 0! = 0.018316$$ $$P(x=1)= 4^1 exp(-4) / 1! = 0.073262$$ $$P(x=2)= 4^2 exp(-4) / 2! = 0.146525$$ $$P(x=3)= 4^3 exp(-4) / 3! = 0.195367$$ $$P(x=4)= 4^4 exp(-4) / 4! = 0.195367$$

P(x<4.5) = 0.018316 + 0.073262 + 0.146525 + 0.195367 + 0.195367

=0.628837

Ans: 0.628837

ii. P(x>3.8)

Solution

P(x>3.8) = 1 - P(x<3.8)

P(x<3.8)= P(x=0) + P(x=1) + P(x=2) + P(x=3)

= 0.018316 + 0.073262 + 0.146525 + 0.195367

P(x<3.8)=0.43347

P(x>3.8)= 1 - 0.43347 = 0.56653

Answer : 0.56653

iii. P(3.8<x<4.5)

Solution

P(3.8<x<4.5) = P(x=4)

= 0.195367

Answer : 0.195367