Question

f X is a normally distributed random variable with a mean of 45 and a standard deviation of 8, find the following probabilities: a. P(X $>$> 50). b. P(X < 32). c. P(37 < X < 48). d. P(X = 45).

Answer 1

$\mu=45 \\ \sigma = 8$

a.

$P(X>50) = 1 -P(X<50) \\ = 1 -P(Z< \frac{50-45}{8}) \\ = 1 -P(Z< 0.625) \\ = 1 -0.7340 \\ = 0.2660$

b.

$P(X<32) = P(Z< \frac{32-45}{8}) \\ = P(Z< -1.625) \\ = 0.0520$

c.

$P(37<X<48) = P(X<48) -P(X<37) \\ = P(Z< \frac{48-45}{8}) -P(Z< \frac{37-45}{8}) \\ = P(Z< 0.375) -P(Z< -1) \\ = 0.6461 -0.1586 \\ = 0.4875$

d.

P(X=45) = 0

Because there is an uncountable infinite number of a value of X, therefore the probability of each individual value is zero.