grandes-ecoles 2015 QII.B.2

grandes-ecoles · France · centrale-maths1__mp Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
Justify the convergence, for any real $q \geqslant 0$, of $\int_q^{+\infty} \frac{r\varphi(r)}{\sqrt{r^2 - q^2}}\,\mathrm{d}r$.

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.

Justify the convergence, for any real $q \geqslant 0$, of $\int_q^{+\infty} \frac{r\varphi(r)}{\sqrt{r^2 - q^2}}\,\mathrm{d}r$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QIII.A QIII.B QIII.C.1 QIII.C.2 QIII.C.3 QIV.A.1 QIV.A.2 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C.1 QIV.C.2 QIV.C.3 QIV.D.1 QIV.D.2 QIV.D.3 QV.A.1 QV.A.2 QV.A.3 QV.B.1 QV.B.2