Answer to Question #273330 in Statistics and Probability for Nis

Question

The amounts of time it takes for a student to complete a statistics quiz uniformly distributed between 30 and 60 minutes. One student is selected at random. Find the probability of the following events.

a) the student requires more than 55 minutes to complete quiz

b) the student completes the quiz in a time between 30 and 40 minutes

c) the student completes the quiz in exactly 37.23 minutes

Answer 1

Given, $X\sim U(a=30, b=60)$

$$f(x)=\dfrac{1}{60-30}, 30\leq X\leq 60$$

$$f(x)=\dfrac{1}{30}, 30\leq X\leq 60$$

a)

$$P(X>55)=\displaystyle\int_{55}^{60}f(x)dx=\displaystyle\int_{55}^{60}\dfrac{1}{30}dx$$

$$=\dfrac{1}{30}[x]\begin{matrix} 60 \\ 55 \end{matrix}=\dfrac{1}{6}$$

b)

$$P(30< X<40)=\displaystyle\int_{30}^{40}f(x)dx=\displaystyle\int_{30}^{40}\dfrac{1}{30}dx$$

$$=\dfrac{1}{30}[x]\begin{matrix} 40 \\ 30 \end{matrix}=\dfrac{1}{3}$$

c)

$$P(X=37.23)=\displaystyle\int_{37.23}^{37.23}f(x)dx=\displaystyle\int_{37.23}^{37.23}\dfrac{1}{30}dx$$

$$=0$$