You have access to the readings of a device that indicates whether a radioactive particle has decayed. However you do not get a continuous reading, you get a reading every second.

(a) A reasonable model for the time the particle takes to decay is that it is a random variable with pdf

f_t~(t) = exp(t), if t 0, or 0 otherwise,

where is a fixed constant. Taking into account that the measurement device rounds up the time and outputs an integer number of seconds (if the time is 0.1 it outputs 1, if it is 13.4 it outputs 14), compute the pmf of the reading from the device. What kind of random variable is this?

(b) What is the pdf of the error between your reading and the true time of decay?