Hydrological Statistics Flashcards
(27 cards)
What is probability?
Measure of how likely an event will occur
Why can actual probability in hydrology only be approximated?
For probability estimate to be accurate the number of observations would be infinite which is impossible
What is a return period?
Average time interval between occurrences of a hydrological event of a given or greater magnitude.
What does a 100-year flood mean?
A 100-year flood will occur on average once in every 100 years
How do you calculate the return period of an event?
i.e. calculate return period of a 2% probability of a flood every year
It is the inverse of the probability of an event.
P = 1 / Return Period
i.e 1/0.02 = 50 year return period
What is intersection probability and its symbol?
A∩B which represents all elements simultaneously in both A and B - intersection of Venn diagram
What is union probability and its symbol?
A∪B which represents all in A or B or both - whole venn diagram
What is the equation for union probability P(A∪B)?
What if A and B are exclusive?
P(A∪B) = P(A) + P(B) - P(A∩B)
P(A∪B) = P(A) + P(B)
What is the equation for the conditional probability of B given A?
What if A and B are independent?
P(B I A) = P(A∩B) / P(A)
P(B I A) = P(B)
What is equation for P(A) if a event B can be split into a set of mutually exclusive and collectively exhaustive events?
P(A) = SUM{ P(A I Bi) * P(Bi) }
When can a Bernoulli distribution be used?
When finding the probability of an event that can only be 2 outcomes
i.e. flooded and not flooded
What is a binomial distribution?
Among n trials of a bernoulli process it gives the the probability of x occurences
How do we solve a binomial distribution problem?
f(x; n, p) = {n x} (p)^x (1-p)^n-x
Use calculator binomial function
How do we find the expected occurrences of a binomial event?
E(X) = np
where n is the amount of trials and p is the probability of event
How does bernoulli and binomial probability highlight a problem?
What does it highlight about the number of occurrences, X, variable?
Although a X-year flood is expected to occur T/X times, when finding the probability of this occurring using a binomial distribution, it is not a likely event.
It highlights that the number of occurrences is a truly random variable.
What is empirical probability?
Nonparametric probability with no theoretical distribution curve
How do we work out empirical probability?
- Rank the data points in descending order from largest to smallest.
- Use equation in data sheet to find the exceedance probability of the mth largest value for each data point
x = total number of data points
m = rank of an individual point - Plot floods and exceedance probability on probability paper (log) - Q against P
- Draw a line of best fit
- Read off corresponding floods and return period (probabilities) from line
Why is sample size important?
Will provide more data points which will make line of best fit more accurate
What is a problem with the California empirical probability formula?
It is biased (over-estimated) and it is impossible to plot the nth data point (100%) on most probability paper.
What is a problem with the Weibull empirical probability formula?
Only the best for uniform distributions
Which empirical formula do we use?
Weibull
What do rainfall statistics involve?
Hence what are the 2 purposes of rainfall statistics?
Rainfall depth and duration
1. Estimate design rainfall depths
2. Assess rarity of observed event
What is a useful diagram for rainfall statistics?
Depth Duration Frequency (DDF)
What are the 2 uses of a DDF diagram and what information is needed for both?
To assess the rarity of observed rainfall events when the duration and rainfall depth information is known or to estimate design rainfall with a predefined return period.
