Question

Let X ∼ B(5, 0.3). Determine:

(b) P(X < 2);

(c) P(X > 2);

Let X ∼ P0(2.1). Determine:

(a) P(X = 5);

(b) P(X < 3);

(c) P(X ≥ 3);

(d) E(X);

(e) var(X).

Answer 1

since X ∼ B(5, 0.3) , this means X is binomial distribution with parameter n=5 and p=0.3 and

for Binomial distribution ,P(X=r)=ⁿC_rp^r(1-p)^n-r  

(b) P(X < 2)=P(X<=1)=0.5282 ( using ms-excel =BINOMDIST(1,5,0.3,1) )

(c) P(X > 2)=1-P(X<=2)=1-0.8369=0.1631

P(X<=2)=0.8369 ( =BINOMDIST(2,5,0.3,1) )

X ∼ P0(2.1), here X is poisson distribution with parameter =2.1 and

for Poisson distribution, P(X=x)=exp(-λ)λ^^x/x!

(a) P(X = 5)= 0.0417 ( =POISSON(5,2.1,0))

(b) P(X < 3)=P(X<=2)= 0.6496 ( =POISSON(2,2.1,1))

(c) P(X ≥ 3)=1-P(X<3)=1-P(X<=2)=1-0.6496=0.3504

(d) E(X)==2.1,

(e) var(X)==2.1

for poisson distribution mean and variance are equal and it is equal to =2.1