Let X denote the number of alpha particles emitted by barium-133 and observed by a Geiger counter in a fixed position. Assume X has a Poisson distribution and is the mean number of counts per second. Let W denote the waiting time to observe 100 counts. There are 25 independent observations that I organized:

.....Frequency...... Frequency

5.3..... 1.... ....6.9..... 2

5.4..... 1.... ....7.0..... 1

5.7..... 1.... ....7.1......2

5.9..... 1.........7.2......1

6.1..... 1........ 7.3..... 2

6.3..... 2........ 7.4..... 1

6.4..... 1........ 7.5..... 1

6.5..... 1........ 7.6..... 3

6.7..... 3

a) Give the p.d.f., mean and variance of W

b) Calculate the sample mean and sample variance of the 25 observations

c) Use the relative frequency of event { } to approximate

For part b, I got a mean of 6.74 and variance of 0.4432. For part c, I got probability of 3.6. I can't figure out how to find the pdf for part a. I hope someone can help me. Thank you in advance.