2  Exercises 2

Continuous Random Variables

Exercise 2.1 Let \(X\) a random variable of density \(f\) defined by: \(f(x) = k x \times {1}_{]0,2[} (x)\).

  1. Calculate \(k\).

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

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

Exercise 2.2 Let \(X\) a random variable of density function:

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

  1. What is the value of \(c\)?

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

Exercise 2.3 Let \(X\) a random variable of density function:

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

  1. Calculate \(k\).

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

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

  4. Let \(Y=X^2\). Determine the probability distribution of \(Y\).

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

Exercise 2.4 Let \(\Phi\) the distribution function of the standard normal distribution.

  1. Let \(X\) a random variable following a standard normal distribution, i.e. \(X \thicksim \mathcal{N}(0,1)\). Using the standard normal distribution table, calculate: \(P(X>2), P(-1<X<1.5)\) et \(P(X<0.5)\).

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

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

Exercise 2.5 A machine produces parts of diameter \(X\) (in cm) which is a random variable that follows a normal distribution of expectation \(\mu\) and variance \(\sigma^2 = (0.01)^2\). What should be the value of \(\mu\) so that the probability that a coin chosen randomly has a diameter greater than 3 cm, is less than 0.01?

Exercise 2.6 (Uniform law and exponential law) Let \(U\) a random variable following a Uniform law on \([0,1]\). Show that \(X= - \ln U\) follows the exponential law.

Exercise 2.7 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. \(B=\int_0^2 e^{-2x^2+4x-2} dx\)

Exercise 2.8 Loi de Laplace

Let \(c > 0\) a constant. Let \(f\) the function defined 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. Calculate the distribution function \(F\) of \(X\).
  3. We admit that \(X\) has an expected value. Calculate it.
  4. What is the probability law of \(Y=|X|\).

Statistic

Exercise 2.9 Is it a random variable?

  1. Mean of the population.
  2. Size of the population.
  3. Sample size.
  4. Average of the sample.
  5. Variance of the sample mean.
  6. Largest value of the sample.
  7. Variance of the population.
  8. Estimated variance of the sample mean.

Central Limit Theorem

Exercise 2.10 With weight problems on the rise in the European population, a new study has began to measure the relationship between weight and the amount of calories consumed per capita. Previous studies show that a European capita consumes an average of \(2700\) calories per day with a standard deviation of \(800\). In this study, we have a sample of \(500\) Europeans.

  1. Define the random variables studied. We name them \(X_i\) and \(\overline{X}_n\).
  2. Use the CLT to give the distribution of the random variable \(\overline{X}_n\).
  3. Calculate the probability that the average calories consumed per day by the Europeans, which we named \(\overline{X}_n\) in the sample, is greater than \(2750\).

Exercise 2.11 Paul and his 9 friends would like to go bowling. They decide to gather their pocket money and hope to get the total amount needed. We can suppose that the pocket money of each one is a random variable \(X_i\) which follows the exponential law of parameter \(\lambda=0.06\). Its density is then \[ f(x)=0.06 e^{-0.06 x} \times 1_{\mathbb{R}^+}(x) \]

Moreover, we admit that \(X_i\) are independent.

  1. Show that the exponential law \(\mathcal{E}(\lambda)\) is an exceptional case of the Gamma law by giving its parameters. (a reminder of the definition of the Gamma distribution is given below).
  2. Let \(S_{10}=\sum_{i=1}^{10} X_{i}\) What is the law of \(S_{10}\)?
  3. Knowing that a game of bowling costs \(15\)€, what is the probability that Paul and his friends can play a game? (Hint: Apply the CLT for \(S_{10}= \sum_{i=1}^n X_i = n \times \overline{X}_n\))
  4. How should \(z>0\) be chosen so that the probability that the total amount of money in the group is greater than \(z\) is equal to \(5 \%\)?
Gamma distribution \(\Gamma(a,\lambda)\)

A random variable \(X\) follows a Gamma distribution of parameters \(a>0\) and \(\lambda>0\), \(X \thicksim \Gamma(a,\lambda)\) if \(X\) has a density function: \[\forall \, x \in \mathbb{R}, \quad f_{a,\lambda} (x)= \frac{\lambda^a}{\Gamma(a)} x^{a-1} e^{-\lambda x} \times {1}_{\mathbb{R}_{+}}(x)\]

  • The Gamma function is defined on \(\mathbb{R}_{+}^*\) by: \(\Gamma(x) = \int_0^{+\infty} t^{x-1}e^{-t} dt\).
  • Properties of the Gamma function:
    • \(\Gamma(x+1)=x\Gamma(x) \quad \forall \, x >0\)
    • \(\Gamma(n+1) = n! \quad \forall n \in \mathbb{N}\)
    • \(\Gamma(1) = 1\) and \(\Gamma(\frac{1}{2})=\sqrt{\pi}\)
  • Properties of the Gamma distribution:
    • \(E(X)= \frac{a}{\lambda}\) and \(V(X)= \frac{a}{\lambda^2}\).
    • If \((X_n)_{n \in \mathbb{N}}\) is a sequence of independent random variables of laws \(\Gamma(a_n,\lambda)\) then the sum random variable \(\sum_{n=1}^N X_n\) also follows a Gamma distribution \(\Gamma(\sum_{n=1}^N a_n, \lambda)\).