Question

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

(a)  P(x≤1),n=5,p=0.3P(x≤1),n=5,p=0.3

P(x)=P(x)=

(b)  P(x>3),n=4,p=0.1P(x>3),n=4,p=0.1

P(x)=P(x)=

(c)  P(x<3),n=7,p=0.7P(x<3),n=7,p=0.7

P(x)=P(x)=

Answer 1

Solution :

Given that ,

a ) p = 0.3

1 - p = 1 - 0.3 = 0.7  

n = 5

P(x ≤1)

Using binomial probability formula ,

P(X x) = ((n! / (n - x)!) * p^x * (1 - p)^{n - x}

P(X 1 ) = (5! / (5 - 1)!) * 0.3¹ * 0.7)^{5 - 1}

p (x   1 ) = p (x = 0 )

+ (5! / (5 - 0)!) * 0.3⁰ * 0.7)^{5 - 1}

p ( x 1 ) = 0.5282

Probability = 0.5282

b ) n = 4

p = 0.1

1 - p = 1 -0.1 = 0.9

(  x > 3 )

p (  x > 3 ) = p (x = 3 )+ p (x = 4)

= (4 / (3 - 4)!) * 0.1³ * 0.9)^{4 +}

  = (4! / (4 - 3)!) * 0.1⁴ * 0.9)³ +

^p (  x >  3 )  = 0.0001

Probability = 0.0001

c ) n = 7

p = 0.7

1 - p = 1 - 0.7 = 0.3

   x < 3 )

p (  x < 3 )  = p (x = 0 )+ p (x = 1) + p (x = 2)

= (7 / (0 - 4)!) * 0.7³ * 0.3)^{4 +}

   = (7! / (1 - 3)!) * 0.7⁴ * 0.3)³ +

  = (7! / (2 - 2)!) * 0.7⁵ * 0.3)²  +

  = (7! / ( 6 - 1)!) * 0.7⁶ * 0.3)¹ +

= (7! / (7 - 0)!) * 0.7⁷ * 0.3)⁰  +

p (  x < 3 )  = 0.2880

Probability = 0.2880