MODULE 6 LECTURE NOTES 1 REVIEW OF PROBABILITY THEORY. Most water resources decision problems face the risk of uncertainty mainly because of the

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1 MODULE 6 LECTURE NOTES REVIEW OF PROBABILITY THEORY INTRODUCTION Most water resources decision problems ace the risk o uncertainty mainly because o the randomness o the variables that inluence the perormance o the systems. The hydrologic variables such as rainall in a command area, inlow to a reservoir, evapo-transpiration o crops which inluence decision making in water resources, are all random variables. Optimization models developed or water resources management must thereore be ormulated to give optimal decisions with an indication o the associated hydrologic uncertainty. Two classical approaches to deal with the hydrologic uncertainty in optimization models are: Implicit Stochastic Optimization (ISO) and Eplicit Stochastic Optimization (ESO). In Implicit Stochastic Optimization (ISO) the hydrologic uncertainty is implicitly incorporated. The optimization model itsel is a deterministic model, in which the hydrologic inputs are varied with a number o equi-probable sequences and the deterministic optimization model is run once with each o the input sequences. Output set is then statistically analyzed to generate a set o optimal decisions. In Eplicit Stochastic Optimization (ESO), the stochastic nature o the inputs is eplicitly included in the optimization model through their probability distributions. Optimization model is a stochastic model and a single run o the model speciies the optimal decisions. Two commonly used ESO techniques are: Chance Constrained Linear Programming (CCLP), and Stochastic Dynamic Programming (SDP). These techniques will be discussed in the ollowing lectures. However, a background o probability theory is essential or ESO, which will be discussed in the present lecture.

2 CONCEPT OF PROBABILITY A sample space S is the area containing all possible outcomes o an eperiment. An event is one subset o these outcomes. Probability is a measure o the likelihood o occurrence o an event. Probability can be assessed in two ways: (i) Objective or posterior probability which is based on the observation o events and (ii) Subjective or prior probability which is based on eperience or judgement. Three basic aioms o probability are: (i) Totality: P(S) = where S is the sample space (ii) Nonnegativity: P(A) where A is an event (iii)mutually eclusive: I A and B are two mutually eclusive events, then P A B = P(A) + P(B) For mutual eclusive events P A B =. Hence, an etension o the above aiom ater relaing mutual eclusiveness will be P A B = P(A) + P(B) - P A B RANDOM VARIABLE A variable whose value is not known or cannot be measured with certainty (or is nondeterministic) is called a random variable (r.v). Eamples o random variables o interest in water resources are rainall, streamlow, time between hydrologic events (e.g. loods o a given magnitude), evaporation rom a reservoir, groundwater levels, re-aeration and deoygenation rates etc. Any unction o a random variable is also a random variable. In this discussion, we use an upper case letter to denote a random variable and the corresponding lower case letter to denote the value that it takes. For eample, daily rainall may be denoted as. The value it takes on a particular day is denoted as. We then associate probabilities with events such as,, etc. Random variable can be essentially classiied into two categories: discrete and continuous. I a r.v. can take on only discrete values,, 3,..., then is a discrete random variable.

3 3 An eample o a discrete random variable is the number o rainy days in a year which may take on values such as,,, 3,... A discrete random variable can assume a inite number o values. On the other hand, i a r.v. can take on all real values in a range, then it is a continuous random variable. Most variables in hydrology are continuous random variables. The number o values that a continuous random variable can assume is ininite. PROBABILITY DISTRIBUTIONS For discrete random variables, the probability distribution is called a probability mass unction and in case o continuous random variables it is called a probability density unction (pd). The cumulative distribution unction (CDF), F(), represents the probability that is less than or equal to, i.e. F() = P( ). The probability mass unction (PMF) o is deined as p() = P( = ). The PMF o a discrete random variable and its CDF (appears as a staircase) are shown in igures (a) and (b) respectively. For a discrete random variable, there are spikes o probability associated with the values that the random variable assumes. For a continuous random variable, the probability density unction (PDF) is deined as df d, where F() is the CDF o. The PDF and the corresponding CDF or continuous random variable are shown in igures (a) and (b) respectively. Probability distributions o continuous random variables are smooth curves. The cumulative distribution unction (CDF) o a continuous random variable denoted by F(), is a non-decreasing unction with a maimum value o. The CDF represents the probability that is less than or equal to, i.e. F() = P( ). Any unction () deined on the real line can be a valid probability density unction i and only i (i) () or all, and (ii) () or all. Given the PMF or PDF, the CDF can be obtained as

4 4 F p or discrete random variables i i n i F d or continuous random variables p() F() F( ) F( ) = p( ) + p( ) 3... N- N 3... N- N Fig. (a) PMF and (b) CDF o a discrete random variable F() () F d F( ) Fig. (a) PDF and (b) CDF o a continuous random variable () Area P a b a b Fig. 3 Probability density unction

5 5 Reerring igure 3, Area under the curve to the let o = a is P( a) Area under the curve to the let o = b is P( b) Area between = a and = b is P[a b]. For a continuous random variable, probability o the random variable taking a value eactly equal to a given value is zero because P d P d d d. d d F().8.5 F - (.5) = F - (.8) = 5 5 Fig. 4 CDF Reerring igure 4, or any given probability α, α, the value o the random variable can be determined rom the CDF as = F - (α). Statistical properties o random variables A population represents the set o all the values taken by a random process. A sample is a subset o the population. The epected value o ( ) r is the r th moment o a random variable about any reerence point =. Mathematically, E r r d or continuous case E N r r i p i or discrete case i

6 6 where E[ ] is a statistical epectation operator. The irst three moments describe the central tendency, variability and asymmetry o the distribution o a random variable. Epected Value or Mean The central tendency is epressed as an epectation as E d or continuous case E N i i p i or discrete case The mean o a r.v is denoted by μ is equal to the epected value, i.e., μ = E[]. Variance It is the second order central moment. The variance o a continuous r.v. is deined as Var E d The positive square root o variance is called the standard deviation, σ. Coeicient o variation is deined as C v. Skewness The asymmetry o PDF o a r.v. is measured by skew coeicient deined as E 3 3 Eample: Probability density unction (PDF) o a random variable is () = 6 = else where

7 7 Determine () Cumulative distribution unction (cd); () Epected value, E(); (3) Variance, Var (); (4) P[.6]; and (5) P[.4.7] Solution:. Cumulative distribution unction F d 6 d 3. Epected value, E() 3. Variance, Var () 6.5 E d d Var d 3 / 6 d. 4. P[.6] P. 6 P. 6 F P[.4.7] P P. 7 F. 7 F P Commonly used probability distributions Three commonly used distributions in water resources are: Normal, Lognormal and Eponential distributions.

8 8 Normal distribution The normal distribution is also called Gaussian distribution. Two parameters are involved in this distribution: mean and variance. A normal random variable with mean μ and variance σ is denoted as ~ N(μ, σ ). The PDF o the normal distribution given by () is epressed as ep or The PDF o normal distribution is bell-shaped and symmetric at = μ as shown in igure 5 () μ The CDF o a normal distribution is Fig. 5 Normal PDF ep d or Normal random variables are usually transormed to standardized variate Z with zero mean and unit variance i.e., Z = ( - μ) / σ. Then PDF o Z can be epressed as z ep z or z Values o (z) obtained by numerical integration are used in the computations or normal distributions.

9 9 Eample: The monthly streamlow at a reservoir site is represented by a random variable which ollows normal distribution with a mean o units and a standard deviation o 5 units. Find () P[ > 5]; () P[ 4] and (3) The low value which will be eceeded with a probability o.8. Solution: The monthly streamlow at a reservoir site is represented by a random variable which () P[ > 5] P 5 P / 5 P Z P Z / 5 () P[ 4] P 4 P P Z. 539 /. 4 / 5 (3) To ind P[ ] =.8 P. 8 P Z / P Z z. 8 P Z z. z / 5 58 units Lognormal distribution This is used when random variable cannot be negative. A r.v. is lognormally distributed i its logarithmic transorm Y=ln() has a normal distribution with mean μ ln and variance σ ln. The PDF o lognormal r.v. is ln ln ep or ln ln

10 Eponential distribution The probability density unction (pd) o an eponential distribution with parameter λ is: e Here λ > is the parameter o the distribution. Mean E[] = / λ Var () = E[ ] - E[] = / λ. The cumulative distribution unction is given by: F d e