Edexcel GCE Statistics 2

Transcription

1 Edexcel GCE Statistics Continuous Random Variables. Edited by: K V Kumaran kumarmaths.weebly.com 1

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10 1. The continuous random variable X has cumulative distribution function F(x) given by F(x) = 0, 1 ( x 7 1, 3 6x 5), x 1 1 x 4 x 4 (a) Find the probability density function f(x). (b) Find the mode of X. (c) Sketch f(x) for all values of x. (d) Find the mean of X. (e) Show that F() > 0.5. (f) Show that the median of X lies between the mode and the mean. Q6, June 001. A continuous random variable X has cumulative distribution function F(x) given by F(x) = 0, kx 8k, kx, x 0, 0 x, x. (a) Show that k = 81. (b) Find the median of X. (c) Find the probability density function f(x). (d) Sketch f(x) for all values of x. (e) Write down the mode of X. (f) Find E(X). (g) Comment on the skewness of this distribution. Q7, Jan 00 kumarmaths.weebly.com 10

11 3. The continuous random variable X has probability density function x, 15, f ( x ) 15 4 x, , (a) Sketch f(x) for all values of x. 0 x, x 7, 7 x 10, otherwise. (b) (i) Find expressions for the cumulative distribution function, F(x), for 0 x and for 7 x 10. x (ii) Show that for < x < 7, F(x) = (iii) Specify F(x) for x < 0 and for x > 10. (c) Find P(X 8.). (d) Find, to 3 significant figures, E(X). 4. The continuous random variable X has cumulative distribution function (8) Q7, June 00 0, 1 F( x ) x (4 x 3 1 ), x 0 x 0, x 1, 1. (a) Find P(X > 0.7). (b) Find the probability density function f(x) of X. (c) Calculate E(X) and show that, to 3 decimal places, Var (X) = One measure of skewness is Mean Mode. Standard deviation (d) Evaluate the skewness of the distribution of X. (6) Q4, Jan 003 kumarmaths.weebly.com 11

12 5. A continuous random variable X has probability density function f(x) where f(x) = k( x x 1) 1 x 0, 0, otherwise where k is a positive integer. (a) Show that k = 3. Find (b) E(X), (c) the cumulative distribution function F(x), (d) P(0.3 < X < 0.3). 6. The continuous random variable X has probability density function Q7, June 003 f(x) = kx( 5 x), 0 x 4, 0, otherwise, where k is a constant. (a) Show that k = (b) Find the cumulative distribution function F(x) for all values of x. (c) Evaluate E(X). (d) Find the modal value of X. (e) Verify that the median value of X lies between.3 and.5. (f) Comment on the skewness of X. Justify your answer. Q7, Jan 004 kumarmaths.weebly.com 1

13 7. A random variable X has probability density function given by f(x) = 1, 3 3 8x 45 0,, 0 x 1, 1 x, otherwise. (a) Calculate the mean of X. (b) Specify fully the cumulative distribution function F(x). (c) Find the median of X. (d) Comment on the skewness of the distribution of X. 8. The random variable X has probability density function (5) (7) Q7, June 004 f(x) = k( x 0, 5x 4), 1 x 4, otherwise. (a) Show that k = 9. Find (b) E(X), (c) the mode of X. (d) the cumulative distribution function F(x) for all x. (e) Evaluate P(X.5), (f ) Deduce the value of the median and comment on the shape of the distribution. (5) Q7, Jan 005 kumarmaths.weebly.com 13

14 9. A continuous random variable X has probability density function f(x) where where k is a positive constant. f(x) = 3 k(4x x ), 0, 0 x, otherwise, (a) Show that k = 4 1. Find (b) E(X), (c) the mode of X, (d) the median of X. (e) Comment on the skewness of the distribution. (f ) Sketch f(x). 10. A continuous random variable X has probability density function f(x) where Q6, June 005 f(x) = kx( x ), x 3, 0, otherwise, where k is a positive constant. (a) Show that k = 4 3. Find (b) E(X), (c) the cumulative distribution function F(x). (d) Show that the median value of X lies between.70 and.75. (6) Q5, Jan 006 kumarmaths.weebly.com 14

15 11. The continuous random variable X has probability density function f(x) = 1 x, k 0, 1 x 4, otherwise. (a) Show that k = 1. (b) Specify fully the cumulative distribution function of X. (c) Calculate E(X). (d) Find the value of the median. (e) Write down the mode. (f ) Explain why the distribution is negatively skewed. 1. The continuous random variable X has cumulative distribution function (5) Q6, June 006 0, x 0, 1, x 1. 3 F( x) x x, 0 x 1, (a) Find P(X > 0.3). (b) Verify that the median value of X lies between x = 0.59 and x = (c) Find the probability density function f(x). (d) Evaluate E(X). (e) Find the mode of X. (f) Comment on the skewness of X. Justify your answer. Q7, Jan 007 kumarmaths.weebly.com 15

16 13. The continuous random variable X has probability density function given by f(x) = 1 x 6 1 x 0 0 x 3 3 x 4 otherwise (a) Sketch the probability density function of X. (b) Find the mode of X. (c) Specify fully the cumulative distribution function of X. (d) Using your answer to part (c), find the median of X. Q8, June The continuous random variable Y has cumulative distribution function F(y) given by (7) F(y) = 0 k( y 1 4 y ) y 1 1 y y (a) Show that k = (b) Find P(Y > 1.5). (c) Specify fully the probability density function f(y). Q4, Jan 008 kumarmaths.weebly.com 16

17 15. The continuous random variable X has probability density function f(x) given by f(x) = ( x ) x3 0 otherwise (a) Sketch f(x) for all values of x. (b) Write down the mode of X. Find (c) E(X), (d) the median of X. (e) Comment on the skewness of this distribution. Give a reason for your answer. Q8, Jan A random variable X has probability density function given by where k is a constant. f(x) = 1 kx 0 x 3 0 x 1 1 x otherwise (a) Show that k = 51. (b) Calculate the mean of X. (c) Specify fully the cumulative distribution function F(x). (d) Find the median of X. (e) Comment on the skewness of the distribution of X. (7) Q7, May 008 kumarmaths.weebly.com 17

18 17. The length of a telephone call made to a company is denoted by the continuous random variable T. It is modelled by the probability density function f(t) = kt, 0 t 10 0, otherwise. 1 (a) Show that the value of k is. 50 (b) Find P(T > 6). (c) Calculate an exact value for E(T) and for Var(T). (d) Write down the mode of the distribution of T. It is suggested that the probability density function, f(t), is not a good model for T. (e) Sketch the graph of a more suitable probability density function for T. 18. A random variable X has probability density function given by (5) Q4, Jan 009 f(x) = 8 9 x 9, 1 x 4 0, otherwise. (a) Show that the cumulative distribution function F(x) can be written in the form ax + bx + c, for 1 x 4 where a, b and c are constants. (b) Define fully the cumulative distribution function F(x). (c) Show that the upper quartile of X is.5 and find the lower quartile. Given that the median of X is 1.88, (6) (d) describe the skewness of the distribution. Give a reason for your answer. Q7, Jan 009 kumarmaths.weebly.com 18

19 19. Figure 1 Figure 1 shows a sketch of the probability density function f(x) of the random variable X. The part of the sketch from x = 0 to x = 4 consists of an isosceles triangle with maximum at (, 0.5). (a) Write down E(X). The probability density function f(x) can be written in the following form. f(x) = ax b ax 0 0 x x 4 otherwise (b) Find the values of the constants a and b. (c) Show that σ, the standard deviation of X, is to 3 decimal places. (d) Find the lower quartile of X. (e) State, giving a reason, whether P( σ < X < + σ) is more or less than 0.5 (7) Q7, June 009 kumarmaths.weebly.com 19

20 0. A continuous random variable X has cumulative distribution function F(x) = 0, x, 6 1, x x 4 x 4 (a) Find P(X < 0). (b) Find the probability density function f(x) of X. (c) Write down the name of the distribution of X. (d) Find the mean and the variance of X. (e) Write down the value of P(X = 1). 1. The continuous random variable X has probability density function f(x) given by Q, Jan 010 f(x) = where k is a constant. k( x x ), 0 x 3, 3k, 3 x 4, 0, otherwise. 1 (a) Show that k =. 9 (b) Find the cumulative distribution function F(x). (6) (c) Find the mean of X. (d) Show that the median of X lies between x =.6 and x =.7. Q4, Jan 010 kumarmaths.weebly.com 0

21 . The lifetime, X, in tens of hours, of a battery has a cumulative distribution function F(x) given by F(x) = 0 4 ( x 9 1 x 3) x 1 1 x 1.5 x 1.5 (a) Find the median of X, giving your answer to 3 significant figures. (b) Find, in full, the probability density function of the random variable X. (c) Find P(X 1.) A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern. (d) Find the probability that the lantern will still be working after 1 hours. 3. The random variable Y has probability density function f(y) given by Q4, June 010 f(y) = ky( a y) 0 y 3 0 otherwise where k and a are positive constants. (a) (i) Explain why a 3. (ii) Show that k =. 9( a ) Given that E(Y ) = 1.75, (b) show that a = 4 and write down the value of k. For these values of a and k, (c) sketch the probability density function, (d) write down the mode of Y. (6) (6) Q7, June 010 kumarmaths.weebly.com 1

22 4. A continuous random variable X has the probability density function f(x) shown in Figure 1. Figure 1 (a) Show that f(x) = 4 = 8x for 0 x.0.5 and specify f(x) for all real values of x. (b) Find the cumulative distribution function F(x). (c) Find the median of X. (d) Write down the mode of X. (e) State, with a reason, the skewness of X. 5. Q5, Jan 011 Figure 1 Figure 1 shows a sketch of the probability density function f(x) of the random variable X. For 0 x 3, f(x) is represented by a curve OB with equation f(x) = kx, where k is a constant. kumarmaths.weebly.com

23 For 3 x a, where a is a constant, f(x) is represented by a straight line passing through B and the point (a, 0). For all other values of x, f(x) = 0. Given that the mode of X = the median of X, find (a) the mode, (b) the value of k, (c) the value of a. Without calculating E(X ) and with reference to the skewness of the distribution (d) state, giving your reason, whether E(X )< 3, E(X ) = 3 or E(X ) > 3. Q3, May The continuous random variable X has probability density function given by f(x) = 3 ( x 1)(5 x) x 5, otherwise. (a) Sketch f(x) showing clearly the points where it meets the x-axis. (b) Write down the value of the mean,, of X. (c) Show that E(X ) = 9.8. (d) Find the standard deviation,, of X. kumarmaths.weebly.com 3

24 The cumulative distribution function of X is given by 0 1 F(x) = ( a 15x 9x 3 1 where a is a constant. x 3 ) x 1 1 x 5 x 5 (e) Find the value of a. (f) Show that the lower quartile of X, q 1, lies between.9 and.31. (g) Hence find the upper quartile of X, giving your answer to 1 decimal place. (h) Find, to decimal places, the value of k so that P( k < X < + k ) = A random variable X has probability density function given by 1, 0 x 1, 1 f(x) = x, 1 x k, 0 otherwise, where k is a positive constant. (a) Sketch the graph of f(x). Q7, May 011 (b) Show that k = 1 (1 + 5). (c) Define fully the cumulative distribution function F(x). (d) Find P(0.5 < X < 1.5). (e) Write down the median of X and the mode of X. (f ) Describe the skewness of the distribution of X. Give a reason for your answer. (6) Q6, Jan 01 kumarmaths.weebly.com 4

25 8. The queuing time, X minutes, of a customer at a till of a supermarket has probability density function f(x) = 3 x( k x) x k, otherwise. (a) Show that the value of k is 4. (b) Write down the value of E(X). (c) Calculate Var (X). (d) Find the probability that a randomly chosen customer s queuing time will differ from the mean by at least half a minute. 9. The continuous random variable X has probability density function f(x) given by Q5, May 01 f(x) = x x x 3, 3 x 4, 4 x 10, otherwise. (a) Sketch f(x) for 0 x 10. (b) Find the cumulative distribution function F(x) for all values of x. (c) Find P(X 8). (8) Q7, May 01 kumarmaths.weebly.com 5

26 30. The continuous random variable T is used to model the number of days, t, a mosquito survives after hatching. The probability that the mosquito survives for more than t days is 5, t 0. ( t 15) (a) Show that the cumulative distribution function of T is given by F(t) = 5 1, t 0, ( t 15) 0, otherwise. (b) Find the probability that a randomly selected mosquito will die within 3 days of hatching. (c) Given that a mosquito survives for 3 days, find the probability that it will survive for at least 5 more days. A large number of mosquitoes hatch on the same day. (d) Find the number of days after which only 10% of these mosquitoes are expected to survive. 31. The continuous random variable X has the following probability density function Q5, Jan 013 where a and b are constants. (a) Show that 10a + 5b =. f(x) = a bx, 0 x 5, 0, otherwise. 35 Given that E(X ) =, 1 (b) find a second equation in a and b, (c) hence find the value of a and the value of b. (d) Find, to 3 significant figures, the median of X. (e) Comment on the skewness. Give a reason for your answer. Q7, Jan 013 kumarmaths.weebly.com 6

27 3. The continuous random variable X has a cumulative distribution function where a and b are constants. 0, x 1, 3 x 3x F(x) = ax b, 1 x, , x, (a) Find the value of a and the value of b. 3 (b) Show that f(x) = (x + x ), 1 x. 10 (c) Use integration to find E(X). (d) Show that the lower quartile of X lies between 1.45 and Q5, May The continuous random variable Y has cumulative distribution function F( y) ( y 4 y ky) 4 1 y 0 0 y y where k is a constant. (a) Find the value of k. (b) Find the probability density function of Y, specifying it for all values of y. (c) Find P(Y > 1). Q, May 013_R kumarmaths.weebly.com 7

28 34. The random variable X has probability density function f(x) given by where k is a constant. (a) Show that k = 1 9. k(3 x x ) 0 x 3 f ( x) 0 otherwise (b) Find the mode of X. (c) Use algebraic integration to find E(X). By comparing your answers to parts (b) and (c), (d) describe the skewness of X, giving a reason for your answer. Q4, May 013_R 35. The length of time, in minutes, that a customer queues in a Post Office is a random variable, T, with probability density function where c is a constant. (a) Show that the value of c is f c81 t t t 9 otherwise 486. (b) Show that the cumulative distribution function F(t) is given by F t 0 3 t t t 0 0 t 9 t 9 (c) Find the probability that a customer will queue for longer than 3 minutes. A customer has been queuing for 3 minutes. (d) Find the probability that this customer will be queuing for at least 7 minutes. Three customers are selected at random. (e) Find the probability that exactly of them had to queue for longer than 3 minutes. Q, June 014 kumarmaths.weebly.com 8

29 36. The continuous random variable X has probability density function f(x) given by x 9 f 9 x 0 x 1 1 x 4 x 4 x otherwise (a) Find E(X). (b) Find the cumulative distribution function F(x) for all values of x. (c) Find the median of X. (d) Describe the skewness. Give a reason for your answer. 37. The random variable X has probability density function f(x) given by where k is a constant. (a) Sketch f (x). (b) Write down the mode of X. 3 k 0 x1 f ( x) kx(4 x) 1 x 4 0 otherwise 9 Given that E X 16 (c) describe, giving a reason, the skewness of the distribution. (d) Use integration to find the value of k. (e) Write down the lower quartile of X. 11 Given also that P( X 3) 36 (f) find the exact value of P(X > 3). (6) Q6, June 014 (5) Q4, June 014_R kumarmaths.weebly.com 9

30 38. In an experiment some children were asked to estimate the position of the centre of a circle. The random variable D represents the distance, in centimetres, between the child s estimate and the actual position of the centre of the circle. The cumulative distribution function of D is given by 0 d 0 4 d d F( d) 0 d 16 1 d (a) Find the median of D. (b) Find the mode of D. Justify your answer. (5) The experiment is conducted on 80 children. (c) Find the expected number of children whose estimate is less than 1 cm from the actual centre of the circle. 39. A random variable X has probability density function given by Q6, June 014_R f(x) = kx x 0 x k1 x otherwise where k is a constant. (a) Show that k = 4 1. (b) Write down the mode of X. (c) Specify fully the cumulative distribution function F(x). (d) Find the upper quartile of X. (5) Q5, June 015 kumarmaths.weebly.com 30

31 40. A continuous random variable X has cumulative distribution function F(x) given by 0 x 3 F x k( ax bx x ) x 3 1 x 3 Given that the mode of X is 8 3. (a) show that b = 8, (b) find the value of k. (6) Q4, June The weight, X kg, of staples in a bin full of paper has probability density function f(x) = 9x 3x 10 0 x 0 otherwise Use integration to find (a) E(X), (b) Var (X), (c) P(X > 1.5). Peter raises money by collecting paper and selling it for recycling. A bin full of paper is sold for 50 but if the weight of the staples exceeds 1.5 kg it sells for 5. (d) Find the expected amount of money Peter raises per bin full of paper. Peter could remove all the staples before the paper is sold but the time taken to remove the staples means that Peter will have 0% fewer bins full of paper to sell. (e) Decide whether or not Peter should remove all the staples before selling the bins full of paper. Give a reason for your answer. Q7, June 016 kumarmaths.weebly.com 31

32 4. The lifetime, X, in tens of hours, of a battery is modelled by the probability density function Use algebraic integration to find (a) E(X) (b) P(X >.5) ì ï f (x) = í ï î 1 x(4 - x) x 4 otherwise A radio runs using of these batteries, both of which must be working. Two fully-charged batteries are put into the radio. (c) Find the probability that the radio will be working after 5 hours of use. Given that the radio is working after 16 hours of use, (d) find the probability that the radio will be working after being used for another 9 hours. 43. The continuous random variable X has a probability density function Q3, June 017 where k is a positive constant. f (x) = ì ï í ï ï î k(x - ) k k(6 - x) 0 x 3 3 < x < 5 5 x 6 otherwise (a) Sketch the graph of f (x). (b) Show that the value of k is 1 3 (c) Define fully the cumulative distribution function F(x). (d) Hence find the 90th percentile of the distribution. (e) Find P[E(X) < X < 5.5] (7) Q6, June 017 kumarmaths.weebly.com 3

33 44. A continuous random variable X has cumulative distribution function 0 x 1 1 F( x) ( x 1) 1 x x 6 (a) Find P(X > 4). (b) Write down the value of P(X 4). (c) Find the probability density function of X, specifying it for all values of x. (d) Write down the value of E(X ). (e) Find Var(X). (f) Hence or otherwise find E(3X + 1). IAL Q, Jan The continuous random variable X has probability density function f(x) given by where k and a are constants. Given that E(X ) = 17 1 k( x a) 1 x f( x) 3 k < x 3 0 otherwise (a) find the value of k and the value of a. (b) Write down the mode of X. (8) IAL Q5, Jan 015 kumarmaths.weebly.com 33

34 46. A continuous random variable X has cumulative distribution function (a) Calculate P(X > 4). 0 x 1 x 4 x 4 0 F( x) 1 x 5 4 x x 5 (b) Find the value of a such that P(3 < X < a) = (c) Find the probability density function of X, specifying it for all values of x. IAL Q1, June A random variable X has probability density function x 0 xk 15 1 f ( x) (5 x) k x otherwise (a) Showing your working clearly, find the value of k. (b) Write down the mode of X. k (c) Find P X X k (5) IAL Q7, June 015 kumarmaths.weebly.com 34

35 48. A continuous random variable X has cumulative distribution function 0 x 0 1 x 0 x1 4 F( x) x 1 x d x d (a) Show that d = (b) Find P(X < 1.5) (c) Write down the value of the lower quartile of X (d) Find the median of X (e) Find, to 3 significant figures, the value of k such that P(X > 1.9) = P(X < k) IAL Q4, Jan A continuous random variable X has probability density function where a and b are constants. ax bx 1 x 7 f( x) 0 otherwise (a) Show that 114a + 4b = 1 Given that a = 1 90 (b) use algebraic integration to find E(X) (c) find the cumulative distribution function of X, specifying it for all values of x (d) find P(X > E(X)) (e) use your answer to part (d) to describe the skewness of the distribution. IAL Q6, Jan 016 kumarmaths.weebly.com 35

36 50. The waiting times, in minutes, between flight take-offs at an airport are modelled by the continuous random variable X with probability density function 1 f ( x) 5 0 (a) Write down the name of this distribution. A randomly selected flight takes off at 9 am. x 7 otherwise (b) Find the probability that the next flight takes off before 9.05 am. (c) Find the probability that at least 1 of the next 5 flights has a waiting time of more than 6 minutes. (d) Find the cumulative distribution function of X, for all x. (e) Sketch the cumulative distribution function of X for x 7. On foggy days, an extra minutes is added to each waiting time. (f) Find the mean and variance of the waiting times between flight take-offs on foggy days. IAL Q4, June A continuous random variable X has probability density function Given that the mode is 1, (a) show that a = b. (b) Find the value of a and the value of b. (c) Calculate F(1.5). ax bx 0 x f ( x) 0 otherwise (d) State whether the upper quartile of X is greater than 1.5, equal to 1.5, or less than 1.5. Give a reason for your answer. (5) IAL Q6, June 016 kumarmaths.weebly.com 36

37 5. The lifetime of a particular battery, T hours, is modelled using the cumulative distribution function F(t) = ì0 t < 8 ï 1 í (74t - 5 t + k) 96 8 t 1 ï î1 t > 1 (a) Show that k = 43 (b) Find the probability density function of T, for all values of t. (c) Write down the mode of T. (d) Find the median of T. (e) Find the probability that a randomly selected battery has a lifetime of less than 9 hours. A battery is selected at random. Given that its lifetime is at least 9 hours, (f) find the probability that its lifetime is no more than 11 hours. IAL Q, Oct A continuous random variable X has the probability density function f(x) shown in Figure 1 where m and k are constants. (a) (i) Show that k = 1 8 ìmx 0 x 5 ï f (x) = ík 5 < x 10.5 ï î0 otherwise (ii) Find the value of m (b) Find E(X) (c) Find the interquartile range of X IAL Q4, Oct 016 kumarmaths.weebly.com 37

38 DO NOT WRITE IN THIS A 54. The time, in thousands of hours, that a certain electrical component will last is modelled by the random variable X, with probability density function f (x) = ì 3 64 x (4 - x) ï í ï îï 0 0 x 4 otherwise Using this model, find, by algebraic integration, (a) the mean number of hours that a component will last, (b) the standard deviation of X. f (x) O 4 x Figure 1 Figure 1 shows a sketch of the probability density function of the random variable X. (c) Give a reason why the random variable X might be unsuitable as a model for the time, in thousands of hours, that these electrical components will last. (d) Sketch a probability density function of a more realistic model. IAL Q4, Jan 017 kumarmaths.weebly.com 38

39 55. The continuous random variable X has probability density function f(x) given by ì 1 ï 0 x3 ï ï f (x) = í 1 ï (6 - x) 10 ï ï ï î0 0 x < x 6 otherwise (a) Sketch the graph of f(x) for all values of x. (b) Write down the mode of X. (c) Show that P(X > ) = 0.8. (d) Define fully the cumulative distribution function F(x). Given that P(X < a X > ) = 5 8 (e) find the value of F(a). (f) Hence, or otherwise, find the value of a. Give your answer to 3 significant figures. IAL Q7, Jan The random variable X has probability density function given by f (x) = ìax + b ï3 í x ï îï 0 1 x < 4 4 x 6 otherwise as shown in Figure 1, where a and b are constants. kumarmaths.weebly.com 39

40 HIS AREA DO NOT WRIT f (x) (a) Show that the median of X is 4 O x (b) Find the value of a and the value of b Figure 1 (c) Specify fully the cumulative distribution function of X 57. A call centre records the length of time, T minutes, its customers wait before being connected to an agent. The random variable T has a cumulative distribution function given by (5) (5) IAL Q3, June 017 F (t) = ì0 ï í0.3t t 3 ï î 1 t < 0 0 t 5 t > 5 (a) Find the proportion of customers waiting more than 4 minutes to be connected to an agent. (b) Given that a customer waits more than minutes to be connected to an agent, find the probability that the customer waits more than 4 minutes. (c) Show that the upper quartile lies between.7 and.8 minutes. (d) Find the mean length of time a customer waits to be connected to an agent. IAL Q5, June 017 kumarmaths.weebly.com 40