grandes-ecoles 2015 QIII.B

grandes-ecoles · France · centrale-maths1__mp Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation

We consider a function $f$ belonging to $\mathcal{B}_1$ and we recall that $$\hat{f}(q,\theta) = \int_{-\infty}^{+\infty} f(q\cos\theta - t\sin\theta,\, q\sin\theta + t\cos\theta)\,\mathrm{d}t$$
Justify that for all $q$ and all $\theta$ we have $\hat{f}(-q, \theta+\pi) = \hat{f}(q,\theta)$.

We consider a function $f$ belonging to $\mathcal{B}_1$ and we recall that
$$\hat{f}(q,\theta) = \int_{-\infty}^{+\infty} f(q\cos\theta - t\sin\theta,\, q\sin\theta + t\cos\theta)\,\mathrm{d}t$$

Justify that for all $q$ and all $\theta$ we have $\hat{f}(-q, \theta+\pi) = \hat{f}(q,\theta)$.

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