If x is a binomial random variable compute Px for each of th

If x is a binomial random variable, compute P(x) for each of the following cases:

(a) P(x?2),n=3,p=0.8

P(x)=

(b) P(x>1),n=7,p=0.5

P(x)=

(c) P(x<6),n=7,p=0.2

P(x)=

(d) P(x?2),n=6,p=0.3

P(x)=

Solution

(a) P(x?2),n=3,p=0.8; q = 0.2

P(x) = 1- P(x=3) = 1 - ³C₃(0.8)³(0.2)⁰ = 0.488

(b) P(x>1),n=7,p=0.5; q = 0.5

P(x)= 1- P(x=1) - P(x=0) = 1 - ⁷C₁(0.5)¹(0.5)⁶ - ⁷C₀(0.5)⁰(0.5)⁷ = 1 - 0.0547 - 0.00781 = 0.9375

(c) P(x<6),n=7,p=0.2 ; q = 0.8

P(x)= 1 - P(x=6) - P(x=7) = 1 - ⁷C₆(0.2)⁶(0.8)¹ - ⁷C₇(0.2)⁷(0.8)⁰ = 1 - (5.12*10^-5) - (1.28*10^-5) = 0.999936

(d) P(x?2),n=6,p=0.3 ; q = 0.7

P(x)= 1 - P(x=0) - P(x=1) - P(x=2) = 1 - ⁶C₀(0.3)⁰(0.7)⁶ - ⁶C₁(0.3)¹(0.7)⁵ = 1 - 0.11765 - 0.30253 = 0.57982