Answer :
Answer:
a) 0.6667 = 66.67% probability that the temperature will be above 80°F.
b) 0.3333 = 33.33% probability that the temperature will be between 80°F and 85°F.
c) The expected temperature is of 82.5ºF.
Step-by-step explanation:
Uniform distribution:
The probability of all outcomes between a and b is the same.
Uniform distribution over the interval from 75°F to 90°F.
This means that [tex]a = 75, b = 90[/tex]
a. What is the probability that the temperature will be above 80°F?
We have that, on the uniform distribution:
[tex]P(X > x) = \frac{b - x}{b - a}[/tex]
In this question:
[tex]P(X > 80) = \frac{90 - 80}{90 - 75} = \frac{10}{15} = 0.6667[/tex]
0.6667 = 66.67% probability that the temperature will be above 80°F.
b. What is the probability that the temperature will be between 80°F and 85°F?
We have that:
[tex]P(c \leq X \leq d) = \frac{d - c}{b - a}[/tex]
Then, in this question:
[tex]P(80 \leq X \leq 85) = \frac{85 - 80}{90 - 75} = \frac{5}{15} = 0.3333[/tex]
0.3333 = 33.33% probability that the temperature will be between 80°F and 85°F.
c. What is the expected temperature?
The expected value of the uniform distribution is:
[tex]E = \frac{a + b}{2}[/tex]
In this question:
[tex]E = \frac{75 + 90}{2} = 82.5[/tex]
The expected temperature is of 82.5ºF.