# Distribution

Classified in Mathematics

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**Q1. (a) Describe
The following terms;**

** i. Aleatory uncertainty**

** ii. Epistemic uncertainty**

**Answer:**

**i. Aleatory
Uncertainty:**

Aleatory (Random or Objective) uncertainty is also called irreducible or inherent Uncertainty.

The word aleatory Derives from the Latin alea, which means the rolling of dice. Thus, an Aleatoric uncertainty is one that is presumed to be the intrinsic randomness of A phenomenon. Interestingly, the word is also used in the context of music, Film and other arts, where a randomness or improvisation in the performance is Implied.

**ii. Epistemic uncertainty:**

Epistemic (Subjective) uncertainty is a Reducible uncertainty that stems from lack of knowledge and data. The word Epistemic derives from the Greek (episteme), which means knowledge. Thus, an Epistemic uncertainty is one that is presumed as being caused by lack of Knowledge (or data). Since Epistemic uncertainty is viewed as reducible as more Information is gathered based on past experience or expert judgment, it requires More attention and careful judgment.

**(b) Factor of safety is used to maintain some degree of safety in
Structural design. Explain the concept of safety factor using load-resistance
Distribution diagram.**

**Answer: **

Traditional design processes do not directly account for the **random** nature
Of most input parameters hence the factor of **safety** is used to maintain some
Degree of safety in structural design. Generally, the factor of safety is
Understood to be the ratio of

The expected strength of response to the expected load.

In practices, both the strength and load are variables, the values of Which are scattered about their respective mean values. When the scatter in the Variables is considered, the factor of safety could potentially be less than unity, And traditional factor of safety based design would fail.

The Figures below effectively explains the concept of factor of safety in Structural design.

**Figure 1
(A): No Factor of safety**

It Can be seen in part (A) of Figure 1 that the factor of safety is zero and hence The greater probability of failure as indicated by the large overlapped area.

**Figure 1
(B): Large Factor of safety**

Part (B) describes the scenario in case of a larger factor of safety and hence very Less chances of failure occurring as there is no overlapping.

**Figure 1
(C): Optimum Factor of safety**

Part (C) depicts the case in which an optimum factor of safety is used. The factor Of safety used is comparatively smaller than as used in part (B) and hence There are few but acceptable chances of failure.

** (c)Reliability assessment techniques is utilized as an initial
Guidance for robust designs of deterministic and probabilistic systems. Compare and explain the differences between deterministic and probabilistic systems.**

**Answer**

**Deterministic
System:**

1.Deterministic systems are perfectly predictable. That is, they follow an
Entirely known rule (law, equation or fixed procedure) so that the **state** of
Each component and of the entire **system** can be given at any time for any time
In the past and future.

2.The states of deterministic systems can be described by statements or by Numbers specifying, for example, physical characteristics of the system (observables, such as length and mass of a physical object).

3.Therefore, a deterministic system is one in which the occurrence of all Events is known with certainty. If the description of the system state at a Particular point of time of its operation is given, the next state can be Perfectly predicted.

4.In mathematics and physics, a deterministic system is a system in which No randomness is involved in the development of future states of the system.

5.A deterministic model will thus always produce the same output from a Given starting condition or initial state.

**Probabilistic
Systems:**

1.Probabilistic systems involve some degree of uncertainty in predicting Their behaviour and require “random variables” to describe the system’s Components and their interactions.

2.There is no general agreement on what “randomness” of a system actually Means (it could, for example, mean generated by chance mechanism, being Unpredictable, showing a lack of an apparent order, etc.).

3.A probabilistic system is one in which the occurrence of events cannot Be perfectly predicted.

4.Though the behavior of such a system can be described in terms of Probability, a certain degree of error is always attached to the prediction of The behavior of the system.

5.The theory of probability is the only analytical tool available to help Map the unpredictable. By describing the states of probabilistic systems by Probability numbers, it uses past knowledge to predict future states.

**Q2. (a) Describe the following topics;**

**i. Discrete random variables**

**ii. Continuous random variables ***(4 marks)*

**Solution:**

A random **variable** *X *takes
On various values *x *within the range -¥<*x*<¥. A
Random variable is denoted by an uppercase letter, and its particular value is
Represented by a lowercase letter.

Random variable are of two types: Discrete and continuous.

**i. Discrete
Random variables**

If the random variable is allowed
To take only discrete values, *x1,x2,x3….,xn*, it is called a discrete random
Variable.

**ii. Continuous random variables**

On the other hand, if the random variable is permitted to take any real Value within a specified range, it is called a continuous random variable.

**(b) If a large number of observations or data
Record exist, then a frequency diagram or histogram can be drawn. Explain the concept of
Histogram. *** (6 marks)*

If a large number
Of observations or data record exist, then a frequency diagram or histogram can
Be drawn.

A histogram is
Constructed by dividing the range of data into intervals of approximately similar
Size and then constructing a rectangle over each interval with an area
Proportional to the number of observations that fell within the interval.

The histogram is a Useful tool for visualizing characteristics of the data such as the spread in The data and locations.

Normally, the Adequate numbers of bin can be computed using the following equation:

*a*= 1 + 3.3 log_{10 }n

where:

A : number of bin / class

N : number of observation (data)

**(c) If Y=a_{1}X_{1}+a_{2}X_{2},
Where a_{1} and a_{2} are constants, prove that
The variance of Y can be obtained as follows;**

*(10 marks**)*

**Q4. (a)
Describe the following topics; (4 marks)**

**i. Lognormal distribution**

vSome engineering problems depend Only on positive values of variables.

vThe Lognormal **distribution** is
The most suitable distribution to be used where negative random variables are
To be avoided, in addition, positive values can be fitted under the Lognormal distribution.

vUnlike the Normal distribution, The Lognormal is unsymmetrical. The domain of the Lognormal distribution lies between 0 to +¥.

vThe mean, median, and mod Values are not the same.

vIn Lognormal distribution, the natural Logarithm of the random variable has a Normal distribution.

• The PDF and CDF Of this distribution function are

l_{x} , x_{x} =
Lognormal parameters.

The Lognormal
Distribution has two parameters, l_{x} and x_{x} which
Can be calculated using information obtained from the mean and standard deviation
Of the Normal distribution.

** ii. Exponential distribution**

vThis distribution is defined
For a positive random variable *x *> *x*o > 0.

vThe Exponential distribution is Often used in reliability engineering as it can represent system having a Constant rate failure.

vThe PDF depends on the constant Parameter, l and is given by

l=
Exponential parameter also known as failure rate.*x _{o}*= an offset, which is assumed to be known a priori (the
Smallest value).

The CDF for the Exponential distribution is defined as:

The mean and Variance are given respectively by:

**Q3. (a) Selection of the distribution function is
An essential part of obtaining probabilistic characteristics of structural
Systems. List three (3) criteria of selection of probability distribution.**

*(3 marks)*

**Selection of
Distribution Criteria**

vThe nature of the problem

vThe underlying assumptions Associated with the distribution

vThe shape of the curve between *fx(x)
*and *Fx(x)* obtained after estimating the data

vThe convenience and simplicity Afforded by the distribution in subsequent computations

**(b) If a cantilever beam
Supports two random loads, F_{1} and F_{2} with
Means and standard deviation (std) of m_{1}=20 kN, s_{1}=4 kN and m_{2}=10 kN, s_{2}=2, the bending moment (M) and the
Shear force (V) at the fixed end due to the two loads are M=L_{1}F_{1}+L_{2}F_{2}
And V=F_{1}+F_{2}respectively. **

*(17 marks)*

i.If two loads are independent, what are the mean and the standard Deviation of the shear and the bending moment at the fix end?

ii.If two random loads are normally distributed, what is the probability That the bending moment will exceed 235 kNm?

iii.If two loads are independent, what is the correlation coefficient between
*V* and *M*?

**Normal Distribution:**

vThe Normal distribution, also known as the Gaussian distribution is
The most widely used distribution in engineering practice due to its simplicity
And convenience, especially a

Theoretical basis of the central **limit** theorem.

vThe central limit theorem states that the sum of many arbitrary distribution Random variables asymptotically follows as normal distribution when the sample Size becomes large.

vThe distribution is often used for small coefficients of variation cases, Such as Young’s modulus, Poisson’s ratio, and other material properties.

vThe parameters that define the Normal distribution are mean value, m*x *and standard deviation, s*x*, denoted as *N (*m*,*s*).*

vOne main and useful criterion of the Normal distribution is it can Be applied to any value of a random variable from -¥ to +¥.

vThe distribution is symmetric (*bell curve*) where the mean, mode
And median values are the same. Mean value can be estimated directly from the
Observed data.

vThe areas under the curve within one, two and three *std *are about
68%, 95.5% and 99.7%.

vAny linear functions of normally distributed random variables are Also normally distributed. A nonlinear function of normally distributed random Variables may or may not be normal.

**Weibull Distribution:**

vThe Weibull distribution has been widely used to solve many engineering Problems.

vIt has been accepted that this distribution is the most useful Density function for reliability estimations and also capable of application to Problems including defining strength of brittle materials, classifying failure Types, scheduling preventive maintenance and inspection activities.

vMoreover, it has become the most important distribution for lifetesting Models such as time to failure for electrical components.

vThe Weibull distribution is very flexible in matching a wide range Of phenomena.

vUnlike the Exponential distribution, the failure rate represented by The Weibull distribution is not constant.

vThe flexibility of the Weibull distribution in representing any Random variable means that it can be identical or similar to the several others Of common distributions.

b=1, identical to the Exponential distribution

b=2, identical to the Rayleigh distribution.

b=2.5, similar to the Lognormal distribution.

b=3.6, similar to the Normal distribution.

b=5, similar to the peaked Normal distribution.

**Limit State:**

vIf, when a structure (or part of a structure) exceeds a specific
Limit, the structure (or part of the structure) is unable to perform as
Required, then the specific limit is

Called a limit state.

vThe structure will be considered unreliable if the failure probability Of the structure limit-state exceeds the required value.

vFor most structures, the limit state can be divided into two Categories: Ultimate limit state and serviceability limit state.

**Ultimate limit state** are
Related to a structural collapse of part or all of the structure. Examples of
The most common ultimate limit states are corrosion, fatigue deterioration, fire,
Plastic mechanism, progressive collapse, fracture etc.

vSuch a limit state should have a very low probability of occurrence, Since it may risk the loss of life and major financial loss.

**Serviceability
Limit state** are related to disruption of the normal
Use of the structures. Examples of serviceability limit state are excessive deflection,
Excessive vibration, drainage, leakage, local damage, etc.

vSince there is less danger than in the case of Ultimate limit state, A higher probability of occurrence may be tolerated in such limit states.

vHowever, people may not use structures that yield too much Deflections, vibrations, etc.

vGenerally, the limit state indicates the margin of safety between The resistance and the load of structures.

vThe limit state function, *g(.)*, and probability of failure *Pf*,
Can

Be defined as

vWhere *R *is the resistance and *S *is the loading of the
System. Both *R(.) *and *S(.) *are functions of random variables *X*.

vThe notation *g(.)<0 *denotes the failure surface region.

vLikewise, *g(.)=0 *and *g(.)>0 *indicate the failure
Surface and safe region respectively.

vThe mean and standard deviation of limit state, *g(.)*, can be
Determined from the elementary definition of mean and variance.

vThe mean of *g(.) *is

vThe safety index indicates the
Distance of the mean of

Safety from g(.)=0.

vThe idea behind the safety index Is that the distance from location measures mg to the limit state surface Provides a good measure of reliability.

vThe distance is measured in units Of the uncertainty scale parameter sg.