Exercises sheet 2

Continuous Random Variables

Exercice 1 Let \(X\) be a continuous random variable of density function \(f\) defined by: \(f(x) = k x \times {1}_{]0,2[} (x)\).

  1. Determine the constant \(k\).

  2. Calculate \(E(X)\) and \(E(X^2)\).

  3. Let \(Z=X^2\). Determine the density function \(Z\). Calculate \(E(Z)\).

Exercice 2 Let \(X\) a continuous random variable of density:

\[f(x)= \left\lbrace \begin{array}{ll} c(1-x^2) & \mbox{if} \quad -1<x<1\\ 0 & \mbox{sinon} \end{array} \right. \]

  1. Calculate \(c\)?

  2. Determine the distribution function of \(X\)?

Exercice 3 Let \(X\) a continuous random variable density:

\[ f(x)= \left\lbrace \begin{array}{ll} 0 & \mbox{if} \quad |x| > k > 0\\ x+1 & \mbox{if} \quad |x| \le k \end{array} \right. \]

  1. Determine \(k\).

  2. Calculate \(E(X)\) and \(E(X^2)\).

  3. Determine the distribution function of \(X\).

  4. Let \(Y=X^2\). Determine the distribution function and the density function of \(Y\).

  5. Calculate \(E(Y)\).

Exercice 4 (Random variable of even density function) Let \(X\) a random variable having an even density function \(f\).

  1. Calculate \(P(X \le 0)\) and \(P(X\ge 0)\).

  2. Show that \(\forall \, x \in \mathbb{R}, F(x)=1-F(-x)\), where \(F\) is the distribution function of \(X\).

  3. Calculate \(E(X)\).

  4. Give an example of an even density function.

Exercice 5 (Laplace distribution) Let \(c > 0\), a real constant. Let \(f\) a function defines on \(\mathbb{R}\) by:

\[ \forall \, x \in \mathbb{R}, \quad f(x) = \frac{c}{2} e^{- c |x|}\]

  1. Show that \(f\) is a density function.
  2. Determine the distribution function \(F\) of \(X\).
  3. Calculate \(E(X)\).
  4. What is the distribution of \(Y=|X|\).

Exercice 6 (Distributions of min(X,Y) and max(X,Y)) Let \(X\) and \(Y\) two random variables of respective density functions \(f_X\) and \(f_Y\). Let \(F_X\) and \(F_Y\) their distribution functions. It is assumed that \(X\) and \(Y\) are independent. Let:

\[Z = max(X,Y) \quad \quad \text{and} \quad \quad T=min(X,Y)\]

  1. Express the distribution functions of \(Z\) and \(T\) with respect to \(F_X\) and \(F_Y\).

  2. Express the density functions of \(Z\) and \(T\) with respect to \(f_X, f_Y, F_X\) and \(F_Y\).

Exercice 7 (Minimum and Maximum of two exponential distributions) Let \(X_1\) and \(X_2\) two independent random variables following the exponential distribution with parameters \(\lambda_1\) and \(\lambda_2\). Let \(X = min(X_1,X_2)\).

  1. Show that \(X\) follows an exponential distribution of parameter \(\lambda_1 + \lambda_2\).

  2. Two counters are open at a bank. The service time at the first counter (resp. at the second) follows an exponential distribution of average 20 min (resp. 30 min). Two customers enter simultaneously, one chooses the counter number 1 and the other the counter number 2.

    1. On average, after how long does the first one come out?

    2. On average, after how long does the last one come out?

Hint: Indication: We can use the relation \(X_1 + X_2 = min(X_1,X_2) + max(X_1,X_2)\). The sum of two real numbers is equal to the sum of their minimum and maximum.

Exercice 8 (Gamma function (Euler)) Gamma function is defined on \(\mathbb{R}_{+}^*\) by:

\[\Gamma(x) = \int_0^{+\infty} t^{x-1}e^{-t} dt\]

  1. Show that \(\Gamma(x+1)=x\Gamma(x)\), \(\forall \, x >0\).

  2. Express \(\Gamma(x+n)\) in function of \(\Gamma(x)\) for \(n\in \mathbb{N}\).

  3. Calculate \(\Gamma(1)\). Deduce \(\Gamma(n+1)\) for \(n\in \mathbb{N}\).

  4. Using the variable change \(t=u^2\), show that: \[\Gamma(x)=2 \int_0^{+\infty} u^{2x-1}e^{-u^2} du\]

  5. Assuming that: \[\int_0^{+\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2}\] Calculate \(\Gamma(\frac{1}{2})\).

Exercice 9 (Loi Gamma) For \(a>0\) and \(\lambda>0\), two real constants, we define the function \(f_{a,\lambda}\) on \(\mathbb{R}\) by: \[\forall \, x \in \mathbb{R}, \quad f_{a,\lambda} (x)= \frac{\lambda^a}{\Gamma(a)} x^{a-1} e^{-\lambda x} \times {1}_{\mathbb{R}_{+}}(x)\]

  1. Show that \(f_{a,\lambda}\) is a density function of \(X\).

  2. Calculate \(E(X)\).

Exercice 10 (Uniform and Exponential distributions) Let \(U\) a random variable of Uniform distribution on \([0,1]\). Show that \(X= - \ln U\) follows an exponential distribution.

Normal (Gaussian) distribution

Exercice 11 Note \(\Phi\) the distribution function of the Standard Normal distribution.

  1. Let \(X\) a random variable following the Standard Normal distribution, i.e. \(X \thicksim \mathcal{N}(0,1)\). Using the table of Standard Normal distribution function, Calculate: \(P(X>2), P(-1<X<1.5)\) and \(P(X<0.5)\).

  2. Let \(Y\) a random variable following a Normal distribution: \(Y \thicksim \mathcal{N}(\mu,\sigma^2)=\mathcal{N}(4,16)\). Calculate: \(P(Y>2), P(-1<Y<1.5)\) and \(P(Y<0.5)\).

  3. Let \(U \thicksim \mathcal{N}(6,4)\). Calculate: \(P(|U-4|<3)\) et \(P( U>6 | U > 3)\).

Exercice 12 A machine produces coins of diameter \(X\) (in cm) that is a random variable following a Normal distribution of expected value \(\mu\) and variance \(\sigma^2 = (0.01)^2\). What should be the value of \(\mu\) so that the probability that a random coin has a diameter greater than 3 cm, is less than 0.01?

Exercice 13 It is planned to build a gatehouse at the entrance of a barrack in which the sentry can shelter in case of bad weather. Sentinels are conscripts whose size is approximately distributed according to a Normal distribution of expected value 175cm and standard deviation of 7cm. At what minimum height must the roof of the gatehouse be located, so that a randomly chosen sentinel has a probability greater than 0.95 of standing there?

Exercice 14 Express the following integrals w.r.t. the distribution function of the standard normal distribution \(\Phi\), then calculate:

  1. \(A=\int_0^1 e^{-\frac{x^2}{2}} dx\)
  2. \(B=\int_0^2 e^{-2x^2+4x-2} dx\)