possesses a moment generating function and the function is called themoment generating functionof . Not all random variables possess a moment generating function. However, all random variables possess acharacteristic function, another transform that enjoys properties similar to those enjoyed by the mgf. ...
Some solved exercises on joint moment generating functions can be found below. Exercise 1Let be a discrete random vector and denote its components by and . Let the support of be and its joint probability mass function be Derive the joint moment generating function of , if it exists. ...
The moment generating function (MGF) of a random variable XX is a function MX(s)MX(s) defined as MX(s)=E[esX].MX(s)=E[esX]. We say that MGF of XX exists, if there exists a positive constant aa such that MX(s)MX(s) is finite for all s∈[−a,a]s∈[−a,a]. ...
The random variable X has moment generating function px(t) = (1 – 0.1t)-10,t<. = 1 0.1 Provide answers to the following to two decimal places (a) Evaluate the natural logarithm of the moment generating function of 4X at the point t= 0.38. HH (b) Hence (o...
Section 4 addresses the generalized case of multi-dimensional problems using a moment-based approach, where more than one random variable may be involved in the limit state function. Illustrative examples are presented in Section 5 to demonstrate the implementation, validity and efficiency of the ...
To this purpose, we exploit the fact that the stiffness matrix is usually a polynomial function of design variables, allowing us to build an equivalent non-linear semidefinite programming formulation over a semi-algebraic feasible set. This formulation is subsequently solved using the Lasserre moment-...
If ψ : ℝ → ℝ + would be continuous, compactly supported and even function, then the problem is reduced to that solved by Lemma 1; ψ can be approximated by dominating even polynomials, the convergence holding uniformly on compact subsets of ℝ . If ψ is not even, while the ...
The system of Equation (14) cannot be solved analytically, so it is not possible to make predictions about the development of vL/vN or m = L/N with respect to the different distribution functions before a model run. During a model run, the ratio vL/vN takes a wide range of values, ...
In principle, the budget equation can be solved with bin modeling. In the case of sedimentation as the only process, however, we find that an analytical solution is possible: 𝑓(𝑡,𝑧,𝐷)=𝑓(0,𝑧+𝑣T(𝐷)𝑡,𝐷)f(t,z,D)=f(0,z+vT(D)t,D) (18) see, for ...