Question

Suppose x is a normally distributed random variable with muμequals=11 and sigmaσequals=2.Find each of the following probabilities.

a.

​ P(xgreater than or equals≥11.511.5​)

  b.

​ P(xless than or equals≤88​)

  c.

​ P(11.6411.64less than or equals≤xless than or equals≤16.216.2​)

  d.

​ P(6.486.48less than or equals≤xless than or equals≤14.114.1​)

Answer 1

Given,

= 11, = 2

We convert this to standard normal as

P( X < x) = p( Z < x - / )

a)

P( X >= 11.5) = P( Z >= 11.5 - 11 / 2)

= P( Z >= 0.25 )

= 0.4013

b)

P( X <= 8) = P( Z <= 8 - 11 / 2)

= P( Z <= 1.5)

= 0.9332

c)

P(11.6 <= X <= 16.2) = P( X <= 16.2 ) - P( X <= 11.6)

= P( Z <= 16.2 - 11 / 2) - P( Z <= 11.6 - 11 / 2)

= P( Z <= 2.6) - P( Z <= 0.3)

= 0.9953 - 0.6179

= 0.3774

d)

P(6.48 <= X <= 14.1) = P( X <= 14.1 ) - P( X <= 6.48)

= P( Z <= 14.1 - 11 / 2) - P( Z <= 6.48 - 11 / 2)

= P( Z < 1.55) - P( Z < -2.26)

= 0.9394 - 0.0119

= 0.9275