Identifying the Probability distribution. If I am certain to receive a phone call in a span of 60 minutes, what distribution does the phone call follow with time, within the 60 minutes period?
Obviously, all instances of time cannot have same probability because if it did, then it would imply that there are chances of not receiving the call even at the end of 60 minutes period, which contradicts our initial assumption. 
To further corroborate on unequal distribution of probability, the chances of a 30 year old getting married is more than that of 18 year old. 
I believe it follows Poisson distribution. I would appreciate very much if anyone derived the probabilities for this question.  

The Poisson distribution would allow more than one call in a given hour; it would also allow zero calls with non-zero probability.
I would not recommend this to model a call that is absolutely certain to occur within a given 60-minute interval.
It's unclear why marriage rates should have anything to do with the question; 18-year-olds have different lives than 30-year-olds, but your life typically doesn't change much between 9 o'clock and 10 o'clock. 
Personally, I would let $X$ be the number of minutes after the start of the period at which the call is received, with a uniform distribution on $[0,60].$
This will give you a $\frac1{60}$ probability to get the call in any given one-minute interval, but unlike a Poisson process, the probability of a call in one interval is not independent of the probability of a call in any other interval. For example, the probability that the call occurs in the last minute of the hour is $\frac1{60}$ a priori but the probability that the call occurs in the last minute, given that it did not occurs in the previous $59$ minutes, is $1.$
