(ENG)

Sorbonne LU3MA246: Processus et simulation 2019/2020

Université 5 March 2020

Test I

Duration: 1h30. 20 marks.

No documents allowed.

The use of electronic devices is strictly forbidden.

Formal and rigorous redaction is expected from the student. The scoring scale provided is solely indicative and is prone to change. Every exercice can be treated separately.

Exercice I: Theoretical background (5 pts)

LetX be a real, random variable. Denote byF_X its cumulative distribution function (fonction de répartition).

1) Recall the definition of the cumulative distribution function.

2) Letα∈]0,1[. Recall the definition of the quantile q_α of level 1−α.

3) Assume X has a symmetric law. Show that for everyα∈]0,1[, P(−q_α/2 ≤X≤q_α/2)

Exercice II: Estimators of the mean and the variance (5 pts)

Let X1, . . . , Xn be real random variables independent and identically distributed (i.i.d), of mean mand varianceσ².

1) Let ¯X_n= _n^{1 Pn}

i=1

X_i. Calculate the mean and the variance of ¯X_n. 2) LetS_n² = _n−1¹

n

P

i=1

X_i−X¯_n². Calculate the mean ofS_n².

3) Now assumeX₁, . . . , X_nare real random variablesi.i.d following the Gaussian lawN(m, σ²).

What is the law of the random variableZn=

n

P

i=1

Xi−nm

√ nσ² ?

4) Letα∈]0,1[. Give a confidence interval of level 1−α for the mean m, when the varianceσ is known.

Exercice III: Simulation of discrete law (3 pts) Let X=p₀δ0 +. . .+p_nδ_n be a discrete random variable.

1) What property should the reals (p_i)_i∈

J0,nK ∈]0,1[ⁿ⁺¹statisfy in order forXto be well-defined?

2) What are the possible values ofX?

1

3) Now considerU a random variable of uniform law on [0,1]. Determine the law of the variable Y =y01{U <p₀}+y11{p₀≤U <p₀+p1}+. . .+yn1{p₀+···+pn−1≤U <p₀+···+p_n},

where (y_i)_i∈

J0,nK∈Rⁿ⁺¹.

Exercice IV: Simulation of Gaussian variables (7pts)

LetR be a real random variable follows a Rayleigh law of parameterσ, and let φbe a random variable of uniform law on [0,2π]. Assume R and φ are independant, and denote by ρ_R and ρ_φ their respective densities.

Recall that a random variable follows a Rayleigh law R(σ) of parameter σ if its density is the following

ρR(r) = r σ²e^−r

2

2σ21]0,+∞[(r).

Let X=Rcos(φ) andY =Rsin(φ).

1) Recall the expression ofρ_φ.

2) Give the density ρ_(R,φ) of the vector (R, φ).

3) Give the density ρ_(X,Y) of the vector (X, Y).

Help: One could rely on the following function

h(x, y) =







arctan ^y_x, x >0, y >0 arctan ^y_x+ 2π, x >0, y <0 arctan ^y_x+π, x <0, y∈R 4) Give the densities of X and Y. AreX and Y independent variables?

2