Discuss the functions of random variables.

What do you understand by conditional probability? Examine the validity of the following statement
If then events A and Bare independent.

For two events A and prove that P(A U

Explain Bayes' theorem and derive its formula.

Discuss the principle of De Morgan's Law.

What is meant by gamma distribution Why is it important for the reliability of structures

Discuss the load as a stochastic process. Write the various types of loads that act on a structure.

Write down the mathematical expression of probability function of Binomial distributions.

The axial load carrying capacity of a column, is normally distributed with MR 1000 kN and 200 kN. The column is subjected to an axial load, which is normally distributed with IlS 700 kN and 300 kN. Calculate the reliability of the column assuming R and S are independent.

Derive an expression for the probability of failure when S (say action due to wind) follows external (largest) distribution and R (say strength of steel) follows the lognormal distribution. Given

r

FS s 0

Prove that, if a function is defined in the interval of a x b and has a relative minimum at x where a b and if the derivative exists as a finite number at x then 0.

Differentiate between Hasofer's and Lind's methods.

Discuss the formulation of optimization problems. What are the various design variables used in the formulation of optimisation of structure?

Explain the Quasi-Newton method. Write about the steepest descent method also.

Discuss the simplex method. Why is it used in linear programming?

Describe any two methods of computing structural reliability.

Write short notes on any two of the following:

Pattern Search Method

Log-normal Distribution

Discrete Variables

Dual Simplex Method