Question

Let X be a normal random variable with a mean of 50 and a standard deviation of 8. Find the following probabilities:

a) P( X ≥ 52 )

b) P( X < 40 )

c) P(X =40 )

d) P( X > 40 )

e) P( 35 ≤ X ≤ 64 )

f) P( 32 ≤ X ≤ 37 )

Answer 1

$Pr(X < x) = \Phi(\frac{x - m}{\sigma})$

where $m = 50, \sigma = 8,$

$\Phi(x) = Pr(X < x) = \int_{-\infty}^{x} e^{\frac{-t^2}{2}}\frac{dt}{\sqrt{2\pi}}$ is the cumulative probability function of the normal distribution with mean = 0 and standard deviation= 1

a)$Pr(X \geq 52) = 1 - Pr(X < 52) =1 - \Phi(\frac{52 - 50}{8}) = 0.4013$

b) $Pr(X < 40) =\Phi(\frac{40 - 50}{8}) = 0.1056$

c)$Pr(X = 40) =0$ because normal distribution is continious

d)$Pr(X > 40) = 1 - \Phi(\frac{40 - 50}{8}) = 0.8944$

e) $Pr(35 \leq X \leq 64) =\Phi(\frac{64 - 50}{8}) - \Phi(\frac{35 - 50}{8}) = 0.9299$

f)$Pr(32 \leq X \leq 37) =\Phi(\frac{37 - 50}{8}) - \Phi(\frac{32 - 50}{8}) = 0.0393$