10  Mixed Exercises

The following chapter contains mixed exercises on the topics covered. The order of the exercises is random.

  1. Imagine the following three-level system for security checks at airports, which is designed to prevent individuals from bringing prohibited substances on board an aircraft: 1) A 3D scan is conducted. 2) A body search is carried out. 3) Detection dogs are used.

    Assume the following probabilities that a prohibited substance is found on a person carrying it:

    • 3D Scan: 90%
    • Body Search: 93%
    • Detection Dogs: 95%

    Assuming that the events of a prohibited substance being detected at each stage are stochastically independent, calculate the probability for the following event:

    For a person carrying a prohibited substance that is not detected in the 3D scan, the substance is found during the body search or by detection dogs.

    Solution: \(0.9965\)

  2. In country X with a population of 4 million people, the population increases by 3% every year.
    How many years does it take for the country to reach a population of 10 million people?

    Solution: \(31\)

  3. A company produces a cheap technical device that is non-functional with a probability of 11%. During a check, 10 devices are tested for functionality.
    What is the probability that 2 or more defective devices are discovered?

    Solution: \(30.28\%\)

  4. It has been found that in people with Covid-19, an antigen test is positive in 85% of cases and a PCR test is positive in 90% of cases. Additionally, in 81% of cases, both the antigen test and the PCR test yield a positive result.
    In what percentage of cases is at least one of the two tests positive?

    Solution: \(94\%\)

  5. Calculate the product \(\mathbf C=\mathbf A\cdot \mathbf B\) using the following data: \[ \mathbf A= \begin{pmatrix}2&4&4\\2&9&9\\1&4&5 \end{pmatrix},\quad \mathbf B= \begin{pmatrix}4&3&6\\1&8&3\\8&6&9 \end{pmatrix}. \] What is the value of \(\mathbf C_{2 3}\)?

    Solution: \(120\)

  6. In a city, the following numbers \(x_i\) of new SARS-COV2 infections were recorded over 5 consecutive days: \[ \begin{array}{c|rrrrr} \text{Day} & 1 & 2 & 3 & 4 & 5\\ \hline x_i & 68 & 55 & 22 & 17 & 42 \end{array} \] Determine a number \(m\) such that \[ \displaystyle \sum_{i=1}^5(x_i-m)^2\quad\to\quad\text{Minimum}. \]

    Solution: \(40.80\)

  7. An investor compiles a portfolio of three securities, the first of which is a fixed-interest security with an interest rate \(r= 0.02\). The other two securities A and B have returns with expected values \(\mu_1= 0.115\) and \(\mu_2= 0.085\), and variances \(\sigma_1^2= 0.035\) and \(\sigma_2^2= 0.03\). The covariance of the returns is \(\sigma_{xy}= 0.025\). The investor wants to create a portfolio \(P=(a_0,a_1,a_2)\) with expected return \(E\) and variance \(V\). Here, \(a_0,a_1,a_2\) represent the respective proportions of the three securities in the portfolio.

    Which of the following statements are correct?

    1. If \(a_0=0\), then \(E(P)=-0.03 a_2+ 0.115\).

    2. If \(a_0=0\), then \(V(P)=0.015 a_2^2 - 0.02 a_2 + 0.035\).

    3. With the Lagrange method, the expected return can be maximized and the risk can be minimized simultaneously.

    Solution: Only statements 1 and 2 are correct.

  8. Three unknowns \(u, v \text{ and } w\) satisfy the following system of equations: \[ \left( \begin{array}{ccc|r} 1 & 2 & 3 & 101\\ 0 & 2 & 3 & 70\\ 0 & 0 & 3 & 0 \end{array}\right) \] Determine \(u+v+w\).

    Solution: \(66\)

  9. Calculate the point of minimum for the following function: \[f(x) = \ln(x^{4}-108x+100).\]

    Solution: \(3\)

  10. A 30-year-old man takes out a survival insurance policy for a capital sum of 88000 CU (currency units), to be paid out after 15 years. The legally binding interest rate for insurance is 7 percent. His probability of dying within one year is \(q_m=0.0013\).
    What is the annual return of a surviving policyholder from such an insurance contract? (Hint: First calculate the corresponding risk premium.)

    Solution: \(0.0714\)

  11. Calculate \(\int_0^{2} x^2\sqrt{1+x^3} dx\).

    Solution: \(\frac{52}9 \approx 5.78\)

  12. On March 26, 2020, 8164 people died from Covid-19 in country X, and the next day the number rose to 8339. Assuming the daily percentage growth remains unchanged, how many days will it take for the number of deaths to double?

    Solution: \(32.7\) days

  13. Given are the two matrices \[ \mathbf A=\left(\begin{array}{rr} 4 & 3\\ 9 & 7\end{array}\right),\quad \mathbf B=\left(\begin{array}{rr} -6 & 6\\ 18 & 0\end{array}\right),\quad \mathbf I = \left(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right) \] Which of the following statements are correct?

    1. With the above data, the following is true: \[ \mathbf B=2 \mathbf A - 14\mathbf I \]

  14. With the above information, for an arbitrary \(2\times 2\) matrix \(\mathbf X\): \[ \mathbf X\mathbf A+\mathbf B\mathbf X=(\mathbf A+\mathbf B)\mathbf X \]

  15. With the above information: \[ \mathbf A(\mathbf B+\mathbf I) = \mathbf B\mathbf A+\mathbf A \]

Solution: 1. and 3.

  1. A company with a quadratic cost function offers a product in a market with perfect competition. The company makes a profit of \(G(x) = -2x^2 +1000x - 25000\) for x units produced per year and this year it produces 250 units. Next year, the market price of the goods decreases by 200 MU while the production cost function remains the same.
    How many units does the company need to produce next year if it wants to maximize its profit next year?

Solution: \(200\)

  1. A die is rolled twice. Let \(X\) denote the number of eyes on the first roll and \(Y\) the number of eyes on the second roll.
    What is \(P(X \leq Y)\)?

Solution: \(0.58\)

  1. To significantly increase the number of tests during the Covid-19 pandemic, the (long known) method of pool testing was proposed: In the specific case, the mucous membrane samples from 25 individuals (after securing the original sample material) are mixed into a common sample and then the PCR test for detection of viral RNA is applied to the common sample. If this test is negative, then a negative test result can be assigned to all 25 subjects. If the test is positive, then the secured samples must be tested person by person. Pool testing can lead to significant efficiency gains regarding the average total number of tests.
    Let \(X\) be the number of PCR tests needed to assign a positive or negative test result to each of the 25 patients.
    Calculate \(E(X)\) if the probability of a positive PCR test in the population is 4.5%. (Hint: \(X\) can only take on two different values.)

Solution: \(18.09\)

  1. For two events \(A\) and \(B\) that can be observed in an experiment, the following contingency table was determined: \[ \displaystyle \begin{array}{l|cc|c} & B & B' &\\ \hline A & 0.3 & 0.1 & 0.4 \\ A'& 0.4 & 0.2 & 0.6\\ \hline &0.7 & 0.3 & 1.0 \end{array} \] Which of the following statements is/are correct?

    1. \(P(A'\cap B) = 0.3\).

    2. \(P(B'|A) = 0.75\).

    3. \(A\) and \(B\) are independent.

Solution: No statement is correct.

  1. An investment is compounded monthly with an effective annual interest rate of 6%.
    What is the nominal interest rate?

Solution: \(5.84\%\)

  1. On an island, a hitherto unknown species of animal is discovered. To estimate the average body size of this new species, a sample of \(n\) randomly selected specimens of this species is collected, their individual body sizes are measured, and the average of these measurements is formed.
    Assuming that the standard deviation of the body size of this species is 7cm, how large must \(n\) be at least so that the standard deviation of the estimate does not exceed 1cm?

Solution: \(49\)

  1. An investor allocates his assets to two stocks whose returns are random variables \(X,Y\) with the same expected return. The variances and covariance are as follows: \[ \sigma_X^2 = 0.36, \quad \sigma_Y^2 = 0.36, \quad \sigma_{X,Y} = 0.1718 \] Find the proportion \(a\) of the assets that must be invested in stock \(X\) to minimize the variance of the portfolio return \(aX + (1-a)Y\).

Solution: \(1/2\)

  1. Calculate the present value of a 10-year annuity, which makes a payment of 1558.15 MU at the end of each month, if the annual effective interest rate is 8%. Round your result to a whole number.

Solution: \(130000\)

  1. Determine the Hessian matrix \[ H(x,y) = \begin{pmatrix}h_{11}(x,y)&h_{12}(x,y)\\h_{21}(x,y)&h_{22}(x,y) \end{pmatrix} \] of \(f(x,y)=\ln(2x^{5y})\) at the point \((x_0,y_0)=(4,3)\).

Which of the following statements is correct?

  1. \(h_{11}(x_0,y_0)=-1.95\)

  2. \(h_{12}(x_0,y_0)=1.25\)

  3. \(h_{22}(x_0,y_0)=0\)

Solution: Only statements 2 and 3 are correct.

  1. A company produces a certain good. Due to delivery fluctuations in the supplier products, the number \(X\) of units of the good produced per day is random.
    The contracts with the company’s customers stipulate that it must deliver at least 1000 units of the good on at least 98 percent of all days.
    What is the expected number of units of the good produced if \(X\) is normally distributed with a standard deviation of 100?

Solution: \(1205.37\)

  1. A fair die is rolled twice. Let \(X\) denote the number of eyes on the first roll and \(Y\) the number of eyes on the second roll.
    Calculate the conditional probability that the sum of the two numbers of eyes is at most 6 given that the same number of eyes has fallen on the first roll as on the second roll. That is, calculate \[P(X+Y\leq 6 | X=Y).\] Round your result to 2 decimal places.

Solution: \(0.50\)

  1. Given is a random variable with its probability function: \[ \small \begin{array}{c|rrrrrrrrrr} X & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\ \hline f(x) & 0.17 & 0.19 & 0.02 & 0.11 & 0.11 & 0.02 & 0.02 & 0.13 & 0.17 & 0.06 \end{array} \] Calculate the probability that \(X\) is an even number.

    Solution: \(0.51\)

  2. A company regularly orders nails from a tool manufacturer and notes that the manufacturer is quite imprecise in the number of nails delivered. For \(N\) nails ordered, the company receives \(X\) nails from the manufacturer instead, where \(X\) is normally distributed with mean \(N\) and variance \(N\).
    How many nails must the company order to get at least 10,000 nails with a 90% probability? (Tip: First calculate \(a\), where \(a = \sqrt{N}\).)

    Solution: \(10129\)

  3. Given are the four matrices \[ \begin{aligned} \mathbf A & = \left(\begin{array}{rr} 3 & 0\\ 0 & 4\\ \end{array}\right),\quad \mathbf B = \left(\begin{array}{rr} -5 & 0\\ 0 & -1\\ \end{array}\right),\quad \mathbf C = \left(\begin{array}{rr} -5 & 0\\ 0 & 1\\ \end{array}\right),\\ \mathbf X & = \left(\begin{array}{rr} 3 & -3\\ 4 & 3\\ \end{array}\right) \end{aligned} \] Calculate \(\mathbf Z=\mathbf A\mathbf X+\mathbf B\mathbf X+\mathbf C\mathbf X\). What is the value of \(z_{21}\)?

    Solution: \(16\)

  4. Given is the following equation matrix: \[ \left(\begin{array}{rrr|r} 1 & 2 & 3 & 3 \\ 0 & 4 & 5 & 5 \end{array}\right) \] Determine the complete solution set of the system of equations.

    Solution: \(\left(\begin{array}{r} -4 \\ -10 \\ 9 \end{array}\right) + \alpha \left(\begin{array}{r} 2 \\ 5 \\ -4 \end{array}\right)\)

  5. Let \(X\) and \(Y\) be random variables with \(\sigma^2_x= 19\), \(\sigma_{xy}= 7\). It is also known that \(V(15X+10Y)=7775\).
    Determine the variance of \(Y\) from this.

    Solution: \(\sigma_y^2=14\)

  6. Calculate \(f_{12}''(v,w)\) for \[ \begin{gathered} f(v,w)=10\cdot\frac{7+v}{8+w} \end{gathered} \] at point \(v=5\) and \(w=5\).

    Solution: \(-0.0592\)

  7. In country XY, there are two ethnic groups A and B. A Covid-19 screening program has found that 12 percent of group A tested positive, while it was 7 percent for group B. Group A makes up 22 percent of the total population.
    Calculate the probability that someone who tested positive belongs to population group A.

    Solution: \(0.3259\)

  8. A quadratic cost function passes through the points \(P_0=(0,0)\), \(P_1=(2,26)\), and \(P_3=(5,140)\).
    Determine the equation of the cost function.

    Solution: \(C(x)=5x^2 +3x\)

  9. A company makes a profit of \(G(p) = 1000\cdot \sqrt{2p-70} \cdot e^{-p/30}\) where \(p\geq 35\) is the number of goods produced per year.
    At what \(p\) does the company maximize its profit?

    Solution: 50

  10. The function \(f(x,y)=x^2+y\) is being optimized under the constraint \(8 x + 10 y=25\). It has a minimum.
    What is the function value at this minimum (rounded to 2 decimal places)?

    Solution: \(2.34\)

  11. A matrix \(\mathbf X = \left( \begin{matrix} a & b \\ 0 & c \end{matrix}\right)\) with unknowns \(a, b, c\geq 0\) satisfies the equation: \[ \mathbf X\mathbf X = \left(\begin{matrix} 16 & 28 \\ 0 & 9 \end{matrix}\right). \] What is \(b\)?

    Solution: \(4\)

  12. An investor creates a portfolio of three securities, of which the first has a fixed interest rate \(r\). The other two securities A and B have expected returns of \(\mu_1\) and \(\mu_2\), and variances \(\sigma_1^2\) and \(\sigma_2^2\). The covariance of the returns is \(\sigma_{xy}\). The investor wants a portfolio with an expected return \(E\) and minimal variance \(V\). The respective shares of the three securities in the portfolio to be optimized are described by \(\alpha_0,\alpha_1,\alpha_2\).

    Which of the following statements are correct?

    1. The Lagrange function to be examined is \[ \begin{array}{rcl} L(\alpha_1,\alpha_2,\lambda) & = & r+\alpha_1(\mu_1-r)+\alpha_2(\mu_2-r)\\ & & -\lambda(\alpha_1^2\sigma_1^2+2\alpha_1\alpha_2\sigma_{xy}+\alpha_2^2\sigma_2^2-V). \end{array} \]

    2. The Lagrange function to be examined is \[ \begin{array}{rcl} L(\alpha_1,\alpha_2,\lambda) & = & \alpha_1^2\sigma_1^2+2\alpha_1\alpha_2\sigma_{xy}+\alpha_2^2\sigma_2^2\\ & & -\lambda(r+\alpha_1(\mu_1-r)+\alpha_2(\mu_2-r)-E). \end{array} \]

    3. It is always required that \(\alpha_2=1-\alpha_1\).

    4. An optimal portfolio solves the system of equations \[ \alpha_1\sigma_1^2+\alpha_2\sigma_{xy}=\frac{\mu_1-r}{2\lambda} \text{ and } \alpha_1\sigma_{xy}+\alpha_2\sigma_{2}^2=\frac{\mu_2-r}{2\lambda}. \]

Solution: only 2.

  1. Given is the linear system of equations: \[ \begin{array}{rrrrrrrr} & x_1 & + & 3x_2 & - & 2x_3 & = & 3\\ - & 6x_1 & - & 4x_2 & + & 18x_3 & = & -5\\ & 3x_1 & + & 2x_2 & - & 9x_3 & = & 3\\ \end{array} \] Find all solutions of the system of equations.

    Solution: no solution

  2. A company has a quadratic cost function of the form \(1000 + 20X + 0.3X^2\), where \(X\) represents the quantity of goods produced.
    Calculate the expected costs of the company under the assumption that \(X\) is a random variable with an expected value of 100 and a standard deviation of 50.

    Solution: \(6750\)

  3. Given are the matrices \[ \mathbf A=\begin{pmatrix} 3& 1\\3& -2\end{pmatrix},\quad \mathbf B=\begin{pmatrix} -45& 27\\-36& -9\end{pmatrix}. \] Solve the matrix equation \(\mathbf A\mathbf X=\mathbf B\) for \(\mathbf X\in\mathbb{R}^{2\times 2}\) and determine the determinant \(\det(\mathbf X)\).

    Solution: \(\det(\mathbf X)=-153\)

  4. In mathematical epidemiology, the function \[I(S)=\dfrac{1}{\sigma}\ln(S)-S+1-\dfrac{1}{\sigma}\ln(0.983)\] plays an important role: here \(I\) is the percentage of people in a population who are infected with a pathogen, \(S\) is the proportion of people who have never been infected and therefore have no immunity against the pathogen. They can get infected through contact with the infected. The parameter \(\sigma\) is the contact number, which is the average number of secondary infections caused by one infected individual.
    Calculate the maximum proportion of infected individuals in the population \(I_{\max}\) when the contact number \(\sigma=1.55\) is.

    Solution: \(8.3\%\)

  5. A fair coin is tossed twice. It is known that at least one of the tosses resulted in heads.
    What is the probability that heads came up in both tosses?

    Solution: \(1/3\)

  6. In how many years will capital double if it earns a nominal interest rate of 5% compounded semiannually for the first half of the time and a nominal interest rate of 7% compounded semiannually for the second half of the time?

    Solution: \(11.73\)

  7. The daily electricity consumption in a certain country is described by a random variable \(X\). This is normally distributed with an expected value of 200 gigawatt-hours and a standard deviation of 20 gigawatt-hours.
    What power plant capacity (in gigawatt-hours per day) must the country maintain in order to cover the electricity consumption without imports on 95% of all days? (Round to the nearest whole number.)

    Solution: \(233\)

  8. A company employs four types of workers:

    • male and full-time employed

    • male and part-time employed

    • female and full-time employed

    • female and part-time employed

    The company has a total of 73 employees. The number of women who are employed full-time is 13, and the total number of male employees is 26. There are 31 employees employed full-time.
    How many women are employed part-time?

    Solution: \(34\)

  9. Find the solution \(\mathbf X\) of the matrix equation \(\mathbf A\mathbf X\mathbf A^{-1}+\mathbf A=\mathbf I\) given \[ \mathbf A=\left(\begin{array}{rr} 4 & -7\\ 5 & -9\\ \end{array}\right) \] What is the value of \(x_{22}\)?

    Solution: \(10\)

  10. Find a primitive function of \(x\sqrt{1+4x^2}\).

    Solution: \(\displaystyle \frac{1}{12}(1+4x^2)^{3/2}+c\)