Chapter II - Statistics I - Exercises

This is a set of exercises created by the teaching Faculty for Statistics I, Chapter II - Random Variables, from the Lisbon Accounting and Business School.

Questions

Question 1

At a Christmas market, a certain stall sells small souvenirs. Let \(X\) a r.v. that counts the number of souvenirs sold in that stall each hour. Let the pdf for \(X\) be:

\(x\) 0 1 2 3 4
\(f(x)\) 0.1 0.2 0.3 0.2 0.2
  1. The probability of selling, within the hour, 3 souvenirs, given that at least 1 was sold is \(0.(2)\).
  2. The events “selling an odd number of souvenirs” and “selling more than 2 souvenirs” are independent.

Question 2

Consider the following random variable \(X\) with its \(pdf\) given by:

\(x\) 0 1 2 3 4
\(f(x)\) 0.05 \(\alpha\) 0.35 .25 .05

  1. \(\alpha\) must equal 0.3.

  2. It must bet the case that \(P(X=2|X\leq 3)=0.632\).

  3. By contrasting the mean and the median of \(X\) we can conclude that the distribution is asymmetric.

Question 3

Let \(X\) be a r.v. with \(cdf\) given by: \[ F(X)=\begin{cases} 0 & x\leq -1 \\ 0.25 & -1\leq x < 0 \\ a & 0 \leq x < 1 \\ 0.6 & 1\leq x < 2 \\ 0.85 & 2 \leq x < 3 \\ 1 & \geq 3 \end{cases} \]

  1. Knowing that \(P(X=0)=0.15\), \(a\) must be equal to

  2. \(P(X>1|0\leq X\leq 3)=0.375\)

  3. By looking to the relative dispersion we can conclude that \(E[X]\) is representative of the balance of the distribution.

Question 4

The number of work absences, in days, of the workers of a company is a random variable with the following pdf

\(x\) 1 2 3 4 5 \(x\notin\{1,2,3,4,5\}\)
\(f(x)\) a 0.15 0.3 .10 b 0

It is known further that 60% of the workers did not skip more than 3 days of work.

  1. The values for a and b are

  2. By looking at the distribution function for \(X\), we can find that the probability that a worker skipped 5 days of work, given that it skipped more than 3 days is equal to 0.75.

  3. After a strike at the company, all workers skipped 1 day of work. The impact on the expected value for skipped days is

Question 5

Let \(X\) a r.v. with the following cdf:

\[ F(x)=\begin{cases} 0 & x < 1 \\ 1/6 & 1\leq x < 2 \\ 1/4 & 2 \leq x < 4 \\ 1/2 & 4 \leq x < 5 \\ 7/12 & 5 \leq x < 6 \\ 1 & x \geq 6 \end{cases} \]

  1. The first and third quartile are:
  2. \(P(X=2)=1/12\)
  3. \(F(3)=0\) Because \(3\) is not a value in the support for \(X\).

Question 6

Let \(X\) a r.v. with the following pdf:

\[ f(x)=\begin{cases} \theta x^2 & 0\leq x < 1 \\ 0 & \mathbb{R}\setminus[0,1] \end{cases} \]

  1. \(\theta\) is equal to
  2. The cdf is given by: \[ F(x)=\begin{cases} 0 & x < 0 \\ x^3 & 0\leq x < 1 \\ 1 & x\geq 1\end{cases} \]
  3. The median for \(X\) is
  4. The mean and standard deviation of \(X\) are \(E[X]=0.75\) and \(V[X]=0.1936\)

Question 7

Given the cdf for the r.v. \(X\): \[ F(x)=\begin{cases} 0 & x <0\\ x/3 & 0\leq x < 3\\ 1 & x\geq 3 \end{cases} \]

  1. \(P(1\leq X \leq 2)=1/3\)
  2. The mean for \(X\) is 2.5

Question 8

Let \(X\) a r.v. with cdf:

\[ F(x)=\begin{cases} 0 & x <1\\ x-1 & 1\leq x < 2\\ 1 & x\geq 2 \end{cases} \]

  1. It is a continuous r.v.
  2. The median is 1.5
  3. \(P(X\leq 1.6 | X > 1.2)=0.4\)

Question 9

Let \(X\) a r.v. absolutely continuous with the following pdf:

\[ F(x)=\begin{cases} 0 & x<8\\ \frac{(x-8)^2}{2} & 8\leq x < 9 \\ 1-\frac{(10-x)^2}{2} & 9\leq x < 10 \\ 1 & x\geq 10 \end{cases} \]

  1. \(P(X\geq 9.2 | 9.5 \leq X \leq 9.5)\) is equal to

Question 10

At SuperStore :convenience_store:, three trained employees are qualified to operate the checkout counters, restock products on the shelves, and perform some administrative tasks. SuperStore has three checkout counters, and at least one of them must always be operating.

At any given day and moment when SuperStore is open to customers, consider the following random variables:

  • \(X\) N of employees in the checkout counters :shopping_cart: :credit_card: .
  • \(Y\) N of employees restocking products on the shelves :package:.

The r.v. \(X\) has \(\Omega_X=\{1,2,3\}\)

  1. If \(F_X\) is the cdf for \(X\), then \(P(2\leq X\leq 2.5)\) might be computed through:
    1. \(F_X(2.5)-F_X(2)\)
    2. \(F_X(2.5)-F_X(2^-)\)
    3. \(F_X(2.5^-)-1+F_x(2)\)
    4. \(F_X(2)-F_X(2.5)\)

For the next questions consider:

\(x\) 1 2 3
\(f_X(x)\) 0.17 0.8 0.03

  1. The distribution function for \(X\) at 2.1 has the value:

  2. \(E[10X-0.6]\) equals

  3. \(E[1/x]\) equals

Consider the following table for the joint probability of \((X,Y)\)

\(X\setminus Y\) 0 1 2
1 \(a\) \(2b\) \(b\)
2 0.1 \(c\) 0
3 0.03 0 0
1
  1. If \(P(X=1|Y=0)=0.05\) then

For the following lines consider \(a=0.02\) and \(b=0.05\) and \(c=0.7\)

  1. \(P(X=2|Y\geq 1)\) is approximately
  2. \(X\) and \(Y\) are independent r.v.s.
  3. Knowing that \(E[Y]=0.9\), \(cov(X,Y)\) equals

Question 11

Consider the following information about the random pair \((X,Y)\):

  • \(X\) has support \(\Omega_X=\{0,1\}\) and \(Y\) has support $_Y={-1,1}
  • \(P(X=0)=0.5\) and \(P(Y=1)=0.6\)
  • \(P(X=1,Y=1)=p\) with \(0<p<1\)
  1. \(p\) must equal to make \(E[XY]=0.1\)
  2. If \(p=0.4\), then \(P(X+Y=0)\) equals
  3. If \(p=0.4\) \(V[X]=0.25\) and \(Cov(X,Y)=0.2\), \(V[2Y-X]\) equals

Question 12

Consider the r.v. \(X\) associated with the distribution function \(F\)

  1. It must be that \(P(X>k)+P(X<k)=1\) \(\forall k\)
  2. If \(a<b\) then \(P(a\leq X\leq b)=F(b)-F(a)+P(X=a)\)
  3. If \(X\) is discrete, with pdf \(f\), then \(F(a)\geq f(a)\) \(\forall a\).

Question 13

Consider the random pair \((X,Y)\) and let \(a,b\) be two scalars (\(\in\mathbb{R}\))

  1. If \(V[X]\) exists, then \(V(X-a)=V(a-X)\)
  2. If \(P(Y=aX+b)=1\) then \(|\rho|=1\)
  3. If \(X\) and \(Y\) are discrete, then it is always true that \(E[XY]=E[X]E[Y]\)