Question

High temperatures in a certain city for the month of August follow a uniform distribution over the interval LaTeX: 61^{\circ}F61 ∘ Fto LaTeX: 91^{\circ}F91 ∘ F. Find the high temperature which 90% of the August days exceed.

Answer 1

Answer:

The required probability for the high temperature which 90% of the August days exceed. is 0.0333

Step-by-step explanation:

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 61° F  to 91° F   . Find the high temperature which 90°  F of the August days exceed.

Let assume that X is the random variable

The probability mass function is:

$f(x) = \dfrac{1}{b-a}$

$f(x) = \dfrac{1}{91-61}$

$f(x) = \dfrac{1}{30}$

Thus; The probability density function of X can be illustrated as :

$f(x) = \left \{ {{ \ \ \dfrac{1}{30}} \atop { \limits }}_ \right. _0$       61 <  x < 91  or otherwise

The required probability for the high temperature at 90°  F can be calculated as follows:

$P(X> 90) = \int\limits^{91}_{90} {f(x)} \, dx$

$P(X> 90) = \int\limits^{91}_{90} \ {\dfrac{1}{30} \, dx$

$P(X> 90) = {\dfrac{1}{30} \int\limits^{91}_{90} \ \, dx$

$P(X> 90) = {\dfrac{1}{30} [x]^{91}_{90}$

$P(X> 90) = {\dfrac{1}{30} (91-90)$

$P(X> 90) = {\dfrac{1}{30} \times 1$

$P(X> 90) = 0.0333$

The required probability for the high temperature which 90% of the August days exceed. is 0.0333