Question

For 80% of lectures, Professor John Smith arrives on time and starts lecturing with delay T. When Professor is late, the starting time delay T is uniformly distributed between 0 and 300 seconds. Find the probability distiribution and probability density function of random variable T.

Answer 1

For continuous uniform distribution probability density function will be:

$f(T) = \begin{cases} 1/(b-a) &\text{for } a\le T\le b \\ 0 &\text{for T<a or T>b} \end{cases}$

$f(T) = \begin{cases} 1/300 &\text{for } 0\le T\le 300 \\ 0 &\text{for } T>300 \end{cases}$

Let's describe probability distribution:

P(0$\le$T$\le$100) = $\int_0^{100}1/300 dx=100/300=1/3$

P(100$\le$T$\le$200) =$\int_{100}^{200}1/300 dx=(200-100)/300=1/3$

P(200$\le$T$\le$300) = $\int_{200}^{300}1/300 dx=(300-200)/300=1/3$

P(T>300) = 0