Question

7. Suppose the number of photons emitted by an atom during a one-minute time window can be modeled as a Poisson random variable with parameter ? = 2. Now, suppose you watch the atom for two minutes, and the number of emitted photons in each minute is an independent Poisson process. Calculate the probability that you see exactly one photon during the two minutes. (Hint: this is the probability that you see no photons in the first minute and one photon in the second minute, plus the probability that you see one photon in the first minute and no photons in the

second minute.) Similarly calculate the probability that you see exactly two photons during the two minutes.

Compare these probabilities to the probabilities that a Poisson random variable with parameter ? = 4 takes the value one, or two. Are they the same?

8. Sketch the PDF and CDF of a continuous random variable that is uniform on [0,2].

9. A line segment of length 1 is cut once at random. What is the probability that the longer piece is more than twice the length of the shorter piece?

10. An atom of Uranium-238 is unstable and will eventually decay (i.e., emit a particle and turn into a different element). Given an atom of Uranium-238, the time elapsed until it decays, in years, is modeled as an Exponential random variable with parameter ? = 0.000000000155. How many years must pass for there to be a 50% chance that the Uranium atom decays?

Answer 1