Exercises
Probability review
Exercise 1.1 Let \((\Omega,\mathcal{A},\mathbb{P})\) be a probability space. Prove that, for each \(B\in\mathcal{A}\) with \(\mathbb{P}(B)>0,\) \(\mathbb{P}(\cdot|B):\mathcal{A}\rightarrow[0,1]\) is a probability function on \((\Omega,\mathcal{A}),\) and hence \((\Omega,\mathcal{A},\mathbb{P}(\cdot|B))\) is a probability space.
Exercise 1.2 Prove that if \(A_1,\ldots,A_k\) are independent events, then
\[\begin{align*} \mathbb{P}\left(\bigcup_{i=1}^kA_i\right)=1-\prod_{i=1}^k(1-\mathbb{P}(A_i)). \end{align*}\]
Exercise 1.3 Prove the law of total probability by expanding \(\mathbb{P}(B)=\mathbb{P}\big(B\cap\big(\cup_{i=1}^k A_i\big)\big)\) (why?).
Exercise 1.4 Prove Bayes’ theorem from the law of total probability and the definition of conditional probability.
Exercise 1.5 Consider the discrete sample space \(\Omega=\{a,b,c,d\}\) and the mapping \(X: \Omega\rightarrow \mathbb{R}\) such that \(X(a)=X(b)=0\) and \(X(c)=X(d)=1.\) Consider the \(\sigma\)-algebra generated by the sets \(\{a\}\) and \(\{c,d\}.\) Prove that \(X\) is a rv for this \(\sigma\)-algebra.
Exercise 1.6 Consider the \(\sigma\)-algebra generated by the subsets \(\{a\}\) and \(\{c\}\) for \(\Omega=\{a,b,c,d\}.\)
- Prove that the mapping in Exercise 1.5 is not a rv for this \(\sigma\)-algebra.
- Define a mapping that is a rv for this \(\sigma\)-algebra.
Exercise 1.7 Consider the experiment consisting in tossing two times a coin.
- Provide the sample space.
- For the \(\sigma\)-algebra \(\mathbb{P}(\Omega),\) consider the rv \(X=\) “Number of heads in two tosses”. Provide the range and the probability function induced by \(X.\)
Exercise 1.8 For the experiment of Exercise 1.7, consider the rv \(Y=\) “Difference between the number of heads and the number of tails”. Obtain its range and its induced probability function.
Exercise 1.9 A dice is rolled two times. Obtain the sample space and the pmf of the following rv’s:
- \(X_1=\) “Sum of the resulting numbers”.
- \(X_2=\) “Absolute value of difference of the resulting numbers”.
- \(X_3=\) “Maximum of the resulting numbers”.
- \(X_4=\) “Minimum of the resulting numbers”.
Indicate, for each case, which is the pre-image by the corresponding rv of the set \([1.5,3.5]\) and which is its induced probability.
Random variables
Exercise 1.10 Assume that the lifespan (in hours) of a fluorescent tube is represented by the continuous rv \(X\) with pdf
\[\begin{align*} f(x)=\begin{cases} c/x^2, & x>100,\\ 0, & x\leq 100. \end{cases} \end{align*}\]
Compute:
- The value of \(c.\)
- The cdf of \(X.\)
- The probability that a tube lasts more than \(500\) hours.
Exercise 1.11 When storing flour in bags of \(100\) kg, a random error \(X\) is made in the measurement of the weight of the bags. The pdf of the error \(X\) is given by
\[\begin{align*} f(x)=\begin{cases} k(1-x^2), & -1<x<1,\\ 0, & \text{otherwise}. \end{cases} \end{align*}\]
- Compute the probability that a bag weighs more than \(99.5\) kg.
- What is the percentage of bags with a weight between \(99.8\) and \(100.2\) kg?
Exercise 1.12 A rv only takes the values \(1\) and \(3\) with a non-zero probability. If its expectation is \(8/3,\) find the probabilities of these two values.
Exercise 1.13 The random number of received calls in a call center during a time interval of \(h\) minutes, \(X_h,\) has a pmf
\[\begin{align*} \mathbb{P}(X_h=n)=\frac{(5h)^n}{n!}e^{-5h}, \quad n=0,1,2,\ldots. \end{align*}\]
- Find the average number of calls received in half an hour.
- What is the expected time that has to pass until an average of \(100\) calls is received?
Exercise 1.14 Let \(X\) be a rv following a Poisson distribution (see Example 1.14). Compute the expectation and variance of the new rv
\[\begin{align*} Y=\begin{cases} 1 & \mathrm{if}\ X=0,\\ 0 & \mathrm{if}\ X\neq 0. \end{cases} \end{align*}\]
Exercise 1.15 Consider a rv with density function
\[\begin{align*} f(x)=\begin{cases} 6x(1-x), & 0<x<1,\\ 0, & \mathrm{otherwise}. \end{cases} \end{align*}\]
Compute:
- \(\mathbb{E}[X]\) and \(\mathbb{V}\mathrm{ar}[X].\)
- \(\mathbb{P}\left(|X-\mathbb{E}[X]|<\sqrt{\mathbb{V}\mathrm{ar}[X]}\right).\)
Moment generating function
Exercise 1.16 For each of the pmf’s given below, find the mgf and, using it, obtain the expectation and variance of the corresponding rv.
- \(\mathbb{P}(X=1)=p,\) \(\mathbb{P}(X=0)=1-p,\) where \(p\in(0,1).\)
- \(\mathbb{P}(X=n)=(1-p)^n p,\) \(n=0,1,2,\ldots,\) where \(p\in(0,1).\)
Exercise 1.17 Compute the mgf of \(\mathcal{U}(0,1).\) Then, use the result to obtain:
- The mgf of \(\mathcal{U}(-1,1).\)
- The mgf of \(\sum_{i=1}^n X_i,\) where \(X_i\sim\mathcal{U}(-1/i,1/i).\)
- \(\mathbb{E}[\mathcal{U}(-1,1)^k],\) \(k\geq 1.\) (You can verify the approach using mgf with direct computation of \(\mathbb{E}[\mathcal{U}(-1,1)^k].\))
Exercise 1.18 Compute the mgf of the following rv’s:
- \(\mathcal{N}(\mu,\sigma^2).\)
- \(\mathrm{Exp}(\lambda).\)
- \(X\sim f(\cdot;\theta)\) given in (1.13).
Exercise 1.19 Compute the mgf of \(X\sim \mathrm{Pois}(\lambda).\)
Exercise 1.20 (Poisson additive property) Prove the additive property of the Poisson distribution, that is, prove that if \(X_i,\) \(i=1,\ldots,n\) are independent rv’s with respective distributions \(\mathrm{Pois}(\lambda_i),\) \(i=1,\ldots,n,\) then
\[\begin{align*} \sum_{i=1}^n X_i\sim \mathrm{Pois}\left(\sum_{i=1}^n\lambda_i\right). \end{align*}\]
Exercise 1.21 (Gamma additive property) Prove the additive property of the gamma distribution, that is, prove that if \(X_i,\) \(i=1,\ldots,n\) are independent rv’s with respective distributions \(\Gamma(\alpha_i,\beta),\) \(i=1,\ldots,n,\) then
\[\begin{align*} \sum_{i=1}^n X_i\sim \Gamma\left(\sum_{i=1}^n\alpha_i,\beta\right). \end{align*}\]
Random vectors
Exercise 1.22 Conclude Example 1.27 by computing the cdf function for all \((x_1,x_2)'\in\mathbb{R}^2.\) Split \(\mathbb{R}^2\) into key regions for doing that.
Exercise 1.23 Consider the random vector \((X,Y)'\) with joint pdf
\[\begin{align*} f(x,y)=e^{-x}, \quad x>0,\ 0<y<x. \end{align*}\]
Compute \(\mathbb{E}[X],\) \(\mathbb{E}[Y],\) and \(\mathbb{C}\mathrm{ov}[X,Y].\)
Exercise 1.24 Obtain the marginal pdf and cdf of \(X_2\) in Example 1.26.
Exercise 1.25 Obtain the marginal cdf and pmf of \(X_1\) in Example 1.27.
Exercise 1.26 Consider the joint pdf of the random vector \((X,Y)'\):
\[\begin{align*} f(x,y)=\begin{cases} cx^{-2}y^{-3},&x>50,\ y>10,\\ 0,&\text{otherwise.} \end{cases} \end{align*}\]
- Find \(c\) such that \(f\) is a pdf.
- Find the marginal pdf of \(X.\)
- Find the marginal cdf of \(X.\)
- Compute \(\mathbb{P}(X>100,Y>10).\)
Exercise 1.27 Consider the joint pdf of the random vector \((X,Y)'\):
\[\begin{align*} f(x,y)=\begin{cases} K(x^2+y^2),&2\leq x\leq 3,\ 2\leq y\leq 3,\\ 0,&\text{otherwise.} \end{cases} \end{align*}\]
- Find the value of \(K\) such that \(f\) is a pdf.
- Find the marginal pdf of \(X.\)
- Compute \(\mathbb{P}(Y>2.5).\)
Exercise 1.28 Obtain the pdfs of \(Y|X=x\) and \(X|Y=y\) in Exercise 1.26 applying (1.4). Check that the conditional pdf integrates one for each possible value \(y.\)
Exercise 1.30 Compute the conditional cdfs of \(Y|X=x\) and \(X|Y=y\) in Exercise 1.26. Do they equal \(y\mapsto F(x,y)/F_X(x)\) and \(x\mapsto F(x,y)/F_Y(y)\)?24
Exercise 1.31 Compute variance-covariance matrix of the random vector with pdf given in Exercise 1.27.
Exercise 1.32 Prove that \(\mathbb{E}[\mathrm{Exp}(\lambda)]=1/\lambda\) and \(\mathbb{V}\mathrm{ar}[\mathrm{Exp}(\lambda)]=1/\lambda^2.\)
Transformations of random vectors
Exercise 1.33 Let \(\boldsymbol{X}\sim f_{\boldsymbol{X}}\) in \(\mathbb{R}^p.\) Show that:
- For \(a\neq0,\) \(a\boldsymbol{X}\sim f_{\boldsymbol{X}}(\cdot/a)/|a|^p.\)
- For a \(p\times p\) rotation matrix25 \(\boldsymbol{R},\) \(\boldsymbol{R}\boldsymbol{X}\sim f_{\boldsymbol{X}}\left(\boldsymbol{R}' \cdot \right).\)
Exercise 1.34 Consider the random vector \((X,Y)'\) with pdf
\[\begin{align} f(x,y)=\begin{cases} K(1 - x^2-y^2),& (x,y)'\in S, \\ 0,&\text{otherwise.} \end{cases} \tag{1.12} \end{align}\]
Obtain:
- The support \(S\) and \(K>0.\)
- The pdf of \(X+Y.\)
- The pdf of \(XY.\)
Exercise 1.35 Let \(X_1,\ldots,X_n\sim\Gamma(1,\theta)\) independent. Using only Corollary 1.2, compute the pdf of:
- \(X_1+X_2.\)
- \(X_1+X_2+X_3.\)
- \(X_1+\cdots+X_n.\)
Exercise 1.36 Corroborate the usefulness of Proposition 1.10 by computing the following expectations as (1) \(\mathbb{E}[g(X)]=\int g(x)f_X(x)\,\mathrm{d}x\) and (2) \(\mathbb{E}[Y]=\int yf_Y(y)\,\mathrm{d}y,\) where \(Y=g(X).\)
- \(X\sim \mathcal{U}(0,1)\) and \(g(x)=x^2.\)
- \(X\sim \mathrm{Exp}(1)\) and \(g(x)=e^x.\)
Exercise 1.37 Consider \(X\sim \mathcal{U}(-2,2)\) and \(Y=g(X),\) with \(g\) as in Example 1.33. Plot the density function \(f_Y\) and explain the intuition of why it has such a “Sauron-inspired” shape. You can validate the form of \(f_Y\) by simulating random values for \(Y\) and drawing its histogram.
Exercise 1.38 Let \(X\sim \mathcal{N}(0,1).\) Compute the pdf of \(X^2.\)
Exercise 1.39 Let \((X,Y)'\sim \mathcal{N}_2(\boldsymbol{0},\boldsymbol{I}_2).\) Compute the pdf of \(X^2+Y^2.\) Then, compute the pdf of \(-2\log U,\) where \(U\sim\mathcal{U}(0,1).\) Is there any connection between both pdfs?
Exercise 1.40 Validate the densities of \(X+Y\) and \(XY\) from Exercise 1.33 with a simulation. To do so:
- Compute the pdf of \(R=\sqrt{1-\sqrt{U}}\) with \(U\sim\mathcal{U}(0,1).\)
- Show that \((X,Y)'=(R\cos\Theta,R\sin\Theta)',\) with \(\Theta\sim\mathcal{U}(0,2\pi)\) independent from \(R,\) actually follows the pdf (1.12).
- Simulate \(X+Y\) and draw its histogram. Overlay \(f_{X+Y}.\)
- Simulate \(XY\) and draw its histogram. Overlay \(f_{XY}.\)
Exercise 1.41 Consider the random variable \(X\) with pdf
\[\begin{align} f(x;\theta)=\frac{\theta}{\pi(\theta^2+x^2)},\quad x\in\mathbb{R},\ \theta>0.\tag{1.13} \end{align}\]
Obtain the pdf of \(Y=X\mod 2\pi.\)