250826 Question A

Question A

1. Name at least 3 properties of a CDF

   $$F_X(x),F:\mathbb{R}\to [0,1]$$

• $F$ is right continuous
• $F$ is non-decreasing
• ${\lim }_{x\to -\infty }{F}_X(x)=0$

---

2. Compute $P(a\le X\le b)$ , $a<b$ and $a,b\in R$ with given CDF and PDF

• $P(a\le X\le e)=F_X(b)-F_X(a)$
• $P(a\le X\le e)={\int }_a^b{f}_X(x)dx$

---

3. When representing a non-normally distributed random vector with the KLE or the gPC expansion, what is the advantage of using a gPC expansion.

---

4. Why are the formulas for Mean and Variance from FOSM just approximations.

   The output model is approximated only with the first order Taylor Approximation

---

5. How does the curse of dimensionality affect the gPC method, if you use the non-intrusive projection method for gPC. Write down the equation of projection

• The gPC coefficients are obtained via $q_i=\frac{\mathbb{E}\left[\mathcal{M}(\cdot ){\Phi }_i(\cdot )\right]}{\mathbb{E}\left[{\Phi }_i(\cdot {)}^2\right]}$
• To calculate the numerator and denominator we use a quadrature method, which has the curse of dimensionality.
• Number of nodes is typically chosen such that the polynomial in the denominator could be calculated exactly.

---

6. What does the accuracy of (non-Intrusive projection) gPC depend on? Name at least two properties

• Polynomial Degree.
• Degree of the chosen quadrature rule.