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:
[tex]f(x) = \dfrac{1}{b-a}[/tex]
[tex]f(x) = \dfrac{1}{91-61}[/tex]
[tex]f(x) = \dfrac{1}{30}[/tex]
Thus; The probability density function of X can be illustrated as :
[tex]f(x) = \left \{ {{ \ \ \dfrac{1}{30}} \atop { \limits }}_ \right. _0[/tex] 61 < x < 91 or otherwise
The required probability for the high temperature at 90° F can be calculated as follows:
[tex]P(X> 90) = \int\limits^{91}_{90} {f(x)} \, dx[/tex]
[tex]P(X> 90) = \int\limits^{91}_{90} \ {\dfrac{1}{30} \, dx[/tex]
[tex]P(X> 90) = {\dfrac{1}{30} \int\limits^{91}_{90} \ \, dx[/tex]
[tex]P(X> 90) = {\dfrac{1}{30} [x]^{91}_{90}[/tex]
[tex]P(X> 90) = {\dfrac{1}{30} (91-90)[/tex]
[tex]P(X> 90) = {\dfrac{1}{30} \times 1[/tex]
[tex]P(X> 90) = 0.0333[/tex]
The required probability for the high temperature which 90% of the August days exceed. is 0.0333
