University of Warwick Introductory Mathematics and Statistics Summer Practice Questions Jeremy Smith Piotr Z. Jelonek Nicholas Jackson This version: 22nd August 2017 Please find below a list of warm-up questions for the Introductory Mathematics and Statistics. These are questions which should help you with revising some of basic mathematical and statistical concepts. Unless the underlying material is explicitly covered in the course these questions do not represent testable material. They are also not indicative of the difficulty level of the pre-sessional lectures. We strongly encourage you to attempt to solve all the questions. 1. Let the joint probability distribution of X and Y be given by: X Y 0 1 2 0 2/8 1/8 1/8 1 2/8 2/8 0/8 Table 1: Joint probability distribution for Q1. (a) What is: (i) Pr (X = 0 Y = 0) (ii) Pr (X = 0 Y = 1) (iii) Pr (X = 1 Y = 2). (b) What is: (i) Pr (X = 0) (ii) Pr (X = 1) (iii) Pr (Y = 0) (iv) Pr (Y = 1) (v) Pr (Y = 2). (c) What is: (i) Pr (X = 0 Y = 0) (ii) Pr (X = 0 Y = 1) (iii) Pr (X = 1 Y = 2). (d) Write out the marginal probability distribution for X. (e) Write out the marginal probability distribution function for Y. (f) Write out the probability distribution function for X + Y. (g) Write out the probability distribution function for (X + Y ) 2. 1
2. A large number of individuals aged 18 to 60 were surveyed regarding introducing voluntary contributions by parents towards funding of primary schools in the state sector. It was found that 24% of individuals aged 18-25 35% of those aged 26-40 and 48% of those aged 41-60 were in favour of voluntary contributions. In the survey 15% of interviewees were aged 18-25 and 45% were aged 26-40. (a) What proportion of all individuals are in favour of voluntary contributions? (b) We know that 54% of the sample was female and that amongst females voluntary contributions have 32% support. In addition for females aged 18-25 the proportion in favour of voluntary contributions is 20% and for those aged 26-40 is 30%. We know that 9% of individuals are female and aged 18-25 and 24% are female and aged 26-40. What is the probability of support for voluntary contributions amongst males? What is the probability a male aged below 41 supports voluntary contributions? 3. For a random sample of 11 manufacturing companies with more than 250 employees the data provided below shows the percentage of the workforce that was female: 36 38 40 41 42 45 46 50 51 52 57. Calculate the following statistics for the percentage of females employed in these companies: (a) Mean (b) Median (c) Standard deviation (d) In fact all companies had a workforce of exactly 250 employees. Given this information calculate the mean and standard deviation for the number of female employees in the companies. (e) You are told that all females in this company are paid 11.50 per hour. Calculate the mean and standard deviation for the hourly wage bill for females in these companies. 4. You are the new lecturer on an introductory economics module and set a test and find that the mean mark is 54 with a standard deviation of 19. The Head of Department (HoD) instructs you that for this module the mean of the test should be 61 and the standard deviation 13. What linear transformation of the data should you apply to ensure you meet the conditions of the HoD? 5. The US Constitution requires senators to be at least 30 years old. Relate the following statements A and B in terms of necessary and sufficient. Statement A: My neighbour is 30 years old. Statement B: My neighbour is a US senator. Briefly explain your answer (do not write more than 3 sentences). 6. An amount of 8000 is invested at 4% per year with annual interest rate payments. (a) What is the balance in the account after three years? (First write down the relevant equation next substitute and compute the precise amount) (b) How long does it take for the balance to be 32000? (Write down the formula next solve it for time it takes and find the exact number) 7. Consider a bond which entitles its bearer to yearly pay-off equal to a. These pay-offs are always paid on the same day of a year and they will be paid forever. Assume that the net risk-free rate r f > 0 is fixed hence the corresponding discount factor is q = 1/(1 + r f ). Find the discounted sum of pay-offs for a = 1000 for the net risk-free rates r f equal to respectively 2% 4% and 5% in two scenarios where the first pay-off: (a) is instantaneous (hence all the payments except the first one are discounted) (b) takes place exactly in a year (all the payments are discounted). 2
8. Find the following limits. You may use L Hôpitals rule whenever you find it convenient. Do not just write down the final answer but also provide a short explanation. Below ln ( ) represents natural logarithm that is log e ( ). 4 x (a) lim 2 x 2 x+2 ( e (d) lim rx 1 rx r 0 r ( ) 2 e ( (g) lim x 1 x x2 2 x 0 3x ). 3x (b) lim 2 27 e x 3 x 3 (e) lim 3x e 2x +x x 0 2 x ) 3 2 (c) lim x 3 x 2 2x 3 x 3 (f) lim x 0 ( 1 ln (x+1) 1 x ) 9. Define a sequence and a limit of a sequence. 10. Find the first derivative with respect to x for: (a) f(x) = 1 2 x4 2x 3 3 2 x2 + 9x where x [ 2 4] (b) f(x) = (x 2 3x) exp ( x 3 ) where x R (the exp notation means the exponential i.e. e x 3 ) (c) f(x) = 2 (d) f(x) = 4x (x 2 +3) ln (2x) exp (2x)+2 where x R where x > 0 (e) f(x) = ln (5cy x ) where c and y are positive constants x > 0 (f) f(x) = (x 3 + x 2 ) 50 where x R (g) f(x) = x 2 + 1 where x R ( ) 1 3 (h) f(x) = where x R x 3. x 1 x+3 11. Define the first derivative of a function f( ) at a point c D where D is function s domain. Use the definition involving h 0. 12. Where is the function: f(x) = e x (1 x) x R increasing and where is it decreasing? Find the candidate for a global maximum of f (the global extreme point of f). Verify it is indeed a maximum. 13. Suppose the demand for a commodity depends on the price per unit P according to the equation D = a bp but that a tax t per unit is imposed on the consumers. The constants a and b are positive. The supply function is S = g(p ) where g (P ) > 0. Then the equilibrium condition is a b(p + t) = g(p ). (a) The equilibrium equation defines P as a differentiable function of t. Find dp/dt and determine its sign. (b) What happens to the price P + t paid by the consumers when t increases? 14. A firm s profit is: π(l) = 6L 2 0.2L 3 where L is the only input representing the number of workers (since workers may work part time L R). (a) Find the number of workers that maximizes the firms average profit per worker φ(l) = π(l)/l by considering the first-order condition. (b) Show that at the optimal value of L in part (a) the marginal product of labour π (L) is equal to the average profit. 3
15. Find the largest value which function: u(x y) = x 2 y defined for x 0 y 0 can attain when 3x + 4y = 72 and find the corresponding value of x and y. 16. The production of a firm is given by: f(k L) = K 0.5 L 0.5 where K 0 L 0. Each worker has a salary of 50000 per year and interest on capital investment is paid at 8% per annum. The firm has 1000000 to spend for salary expenses and interest payment each year. Find the values of K and L that maximise production capacity subject to the budget constraint. 17. A firm produces two commodities A and B. The inverse demand functions are: p A = 900 2x 2y and p B = 1400 2x 4y where the firm produces and sells x units of a commodity A and y units of commodity B. The costs are given by: c A = 7 000 + 100x + x 2 and c B = 10 000 + 6y 2. (a) Show that the firm s profits are given by: π(x y) = 3x 2 10y 2 4xy +800x+1400y 17000. (b) Suppose the firm is required to produce a total of 60 units. Find the values of x and y that maximise profits. 18. Find f(x y)/ t when: (a) f(x y) = e xy3 x = t 2 + s y = t 3 + s 2 and t > 0 s > 0 (b) f(x y) = x 2 h(x y) x = t 2 + s y = t 3 + s 2 and t R s R (above h( ) is some differentiable function). 19. Assume that f( ) g( ) and h( ) are all differentiable functions. If: (a) z = F (x y) = x 2 + e y x = t 3 y = 2t find dz/dt (b) Y = F (K L) = KL 2 K = f(t) L = g(t) find an expression for dy/dt (c) g(r) = F (r 1 r 1/(1 r)) find an expression for g (r) (d) z = F (x y) x = f(t) y = h(t s) find expressions for z/ t z/ s. 20. Solve the following systems of simultaneous equations: { x + 2y = 5 (a) 2x + y = 4 { 2x + 3y = 1 (b) 5x + 2y = 3 x + y + z = 11 (c) 2x y + 3x = 19 3x + 2y z = 2 21. Factorise the following polynomials into linear factors: (a) x 2 4 (b) x 2 5x + 6 (c) x 3 x 2 6x (d) 2x 2 5x + 2 22. Given that x 3 6x 2 + 11x 6 has (x 1) as a factor completely factorise this polynomial into linear factors. 4
23. Recall that the solutions x 1 and x 2 of a quadratic equation ax 2 + bx + c = 0 are given by the formula x = b ± b 2 4ac. (1) 2a Show that x 1 + x 2 = b a and x 1x 2 = c a. 24. Use the quadratic formula (1) to solve the following equations: (a) 4x 2 1 = 0 (b) x 2 + 2x + 1 = 0 (c) x 2 + x + 1 = 0 (d) x 2 = x + 1 25. Solve the matrix equation: K + LQN MQN = P for the n n matrix Q. Assume that K L M N and P are n n matrices such that N and (L M) have inverses. 5