6_Probability Density Functions

IB Questionbank Mathematics Higher Level 3rd edition 1

1. The random variable X has probability density function f where

f(x) =

otherwise.,0

20),2)(1( xxxkx

(a) Sketch the graph of the function. You are not required to find the coordinates of the

maximum. (1)

(b) Find the value of k. (5)

(Total 6 marks)

2. A continuous random variable X has a probability density function given by the function f(x), where

f(x) =

otherwise.,03

40,

02)2( 2

xk

xxk

(a) Find the value of k. (2)

(b) Hence find

(i) the mean of X;

(ii) the median of X. (5)

(Total 7 marks)


3. The ten numbers x1, x2, ..., x10 have a mean of 10 and a standard deviation of 3.

Find the value of

10

1

2)12(i

ix .

(Total 6 marks)

4. A continuous random variable X has probability density function

f(x) =

.0,e

0,0

xa

xax

It is known that P(X < 1) = 1 2

1.

(a) Show that a = 2ln2

1.

(6)

(b) Find the median of X. (5)

(c) Calculate the probability that X < 3 given that X > 1. (9)

(Total 20 marks)

5. A continuous random variable X has the probability density function f given by

f(x) =

otherwise.,0

10),( 2 xxxc


(a) Determine c. (3)

(b) Find E(X). (2)

(Total 5 marks)

6. A discrete random variable X has a probability distribution given in the following table.

x 0.5 1.5 2.5 3.5 4.5 5.5

P(X = x) 0.15 0.21 p q 0.13 0.07

(a) If E(X) = 2.61, determine the value of p and of q. (4)

(b) Calculate Var (X) to three significant figures. (2)

(Total 6 marks)

7. Tim throws two identical fair dice simultaneously. Each die has six faces: two faces numbered 1, two faces numbered 2 and two faces numbered 3. His score is the sum of the two numbers

shown on the dice.

(a) (i) Calculate the probability that Tim obtains a score of 6.

(ii) Calculate the probability that Tim obtains a score of at least 3. (3)

Tim plays a game with his friend Bill, who also has two dice numbered in the same way. Bills score is the sum of the two numbers shown on his dice.

(b) (i) Calculate the probability that Tim and Bill both obtain a score of 6.

(ii) Calculate the probability that Tim and Bill obtain the same score. (4)


(c) Let X denote the largest number shown on the four dice.

(i) Show that P(X 2) = 81

16.

(ii) Copy and complete the following probability distribution table.

x 1 2 3

P(X = x) 81

1

(iii) Calculate E(X) and E(X2) and hence find Var(X).

(10)

(d) Given that X = 3, find the probability that the sum of the numbers shown on the four dice

is 8. (4)

(Total 21 marks)

8. A random variable has a probability density function given by

f(x) =

elsewhere.,0

20),2( xxkx

(a) Show that k = 4

3.

(4)

(b) Find E(X). (2)

(Total 6 marks)


9. The annual weather-related loss of an insurance company is modelled by a random variable X

with probability density function

f(x) =

otherwise.,0

200)200(5.2

5.3

5.2

xx

Find the median. (Total 8 marks)

10. In each round of two different games Ying tosses three fair coins and Mario tosses two fair

coins.

(a) The first game consists of one round. If Ying obtains more heads than Mario, she receives

$5 from Mario. If Mario obtains more heads than Ying, he receives $10 from Ying. If

they obtain the same number of heads, then Mario receives $2 from Ying. Determine Yings expected winnings.

(12)

(b) They now play the second game, where the winner will be the player who obtains the

larger number of heads in a round. If they obtain the same number of heads, they play

another round until there is a winner. Calculate the probability that Ying wins the game. (8)

(Total 20 marks)

11. The random variable T has the probability density function

f (t) = .11,2

cos4

t

t

Find

(a) P(T = 0); (2)


(b) the interquartile range. (5)

(Total 7 marks)

12. The probability distribution of a discrete random variable X is defined by

P(X = x) = cx(5 x), x = 1, 2, 3, 4.

(a) Find the value of c. (3)

(b) Find E(X). (3)

(Total 6 marks)

13. A continuous random variable X has probability density function

f (x) =

otherwise.,0

,10for,112 2 xxx

Find the probability that X lies between the mean and the mode. (Total 6 marks)

14. Anna has a fair cubical die with the numbers 1, 2, 3, 4, 5, 6 respectively on the six faces. When she tosses it, the score is defined as the number on the uppermost face. One day, she decides to

toss the die repeatedly until all the possible scores have occurred at least once.

(a) Having thrown the die once, she lets X2 denote the number of additional throws required

to obtain a different number from the one obtained on the first throw. State the

distribution of X2 and hence find E(X2).

(3)


(b) She then lets X3 denote the number of additional throws required to obtain a different

number from the two numbers already obtained. State the distribution of X3 and hence

find E(X3).

(2)

(c) By continuing the process, show that the expected number of tosses needed to obtain all

six possible scores is 14.7. (5)

(Total 10 marks)

15. John removes the labels from three cans of tomato soup and two cans of chicken soup in order to enter a competition, and puts the cans away. He then discovers that the cans are identical, so

that he cannot distinguish between cans of tomato soup and chicken soup. Some weeks later he

decides to have a can of chicken soup for lunch. He opens the cans at random until he opens a can of chicken soup. Let Y denote the number of cans he opens.

Find

(a) the possible values of Y, (1)

(b) the probability of each of these values of Y, (4)

(c) the expected value of Y. (2)

(Total 7 marks)

16. A continuous random variable X has a probability density function given by

f(x) =

otherwise.,0

,3 1for ,60

)1( 3x

x

Find

(a) P(1.5 X 2.5); (2)


(b) E(X); (2)

(c) the median of X. (3)

(Total 7 marks)

17. A discrete random variable X has its probability distribution given by

P(X = x) = k(x + 1), where x is 0, 1, 2, 3, 4.

(a) Show that k = 15

1.

(3)

(b) Find E(X). (3)

(Total 6 marks)

18. The probability density function of the random variable X is given by

f(x) =

otherwise.,0

10for ,4 2

xx

k

(a) Find the value of the constant k. (5)

(b) Show that E(X) =

)32(6 .

(7)


(c) Determine whether the median of X is less than 2

1 or greater than

2

1.

(8)

(Total 20 marks)

19. The continuous random variable X has probability density function

f(x) = 6

1x(1 + x

2) for 0 x 2,

f(x) = 0 otherwise.

(a) Sketch the graph of f for 0 x 2. (2)

(b) Write down the mode of X. (1)

(c) Find the mean of X. (4)

(d) Find the median of X. (5)

(Total 12 marks)

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6_Probability Density Functions