Awhole number is less than 65. if the number is dividedby 9, the remainder is 6. if thenumber is divided by 8, the remainder is 4. what is the whole number?

The whole numbers which give the remainder x when divided by y can be found using the formula yz+x, where z is 0, 1, 2, ...

The whole numbers less than 65 which give the remainder 6 when divided by 9 are:
6, 15, 24, 33, 42, 51, 60

The whole numbers less than 65 which give the remainder 4 when divided by 8 are:
4, 12, 20, 28, 36, 44, 52, 60

The only number that satisfies both these conditions is 60.
The number is 60.

a) What is the probability that the duration of a particular rainfall event at this location is at least 2 hours?

We want this probability"

At most 3 hours?

b) What is the probability that rainfall duration exceeds the mean value by more than 2 standard deviations?

What is the probability that it is less than the mean value by more than one standard deviation?

Step-by-step explanation:

Previous concepts

The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution". The probability density function is given by:

The cumulative distribution for this function is given by:

We know the value for the mean on this case we have that :

Solution to the problem

Part a

What is the probability that the duration of a particular rainfall event at this location is at least 2 hours?

We want this probability"

At most 3 hours?

Part b

What is the probability that rainfall duration exceeds the mean value by more than 2 standard deviations?

The variance for the esponential distribution is given by:

And the deviation would be:

And the mean is given by

Two deviations correspond to 5.540, so we want this probability:

What is the probability that it is less than the mean value by more than one standard deviation?

For this case we want this probablity: