A Level Further Mathematics for AQA Statistics Student Book 10 A fair dice is thrown until a 6 has been thrown or three throws have been made. T is the discrete random variable number of throws taken. a Write down, in tabular form, the distribution of T . b Find E(T ). c Find the median of T. d The number of points awarded in the game, P , is given by 4 − T . Find the variance of P . 11 a A four-sided dice labelled with the values 1 to 4 is rolled twice. Write down, in a table, the probability distribution of S, the sum of the two rolls. b Find E(S ) and Var(S ) .
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c A four-sided dice is rolled once and the score, X , is twice the result. Find the mean and variance of X .
12 The discrete random variable X follows the U(9) distribution. µ is the expectation of X and σ 2 is the variance of X . Find P( µ − σ < X < µ + σ )
Find, in terms of n:
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13 X is a discrete random variable satisfying P ( X = x ) = kx for x = 1, 2, 3, ..., n.
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d E 1 X 14 A box contains a large number of pea pods. The number of peas in a pod can be modelled by the random variable X . The probability distribution of X is shown here: b E( X )
a k
c Var( X )
x
2 or fewer
3
4
5
6
7
8 or more
P( X = x )
0
0.1
0.2
a
0.3
b
0
a Two pods are picked randomly from the box. Find the probability that the number of peas in each pod is at most 4.
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b It is given that E( X ) = 5.1.
i Determine the values of a and b.
ii Hence show that Var( X ) = 1.29.
iii Some children play a game with the pods, randomly picking a pod and scoring points depending on the number of peas in the pod. For each pod picked, the number of points scored, N, is found by doubling the number of peas in the pod and then subtracting 5. Find the mean and the standard deviation of N. [© AQA 2014]
15 A random variable has E( X n ) = n. Find Var( X n ) in terms of n. 16 I n a card game a pack of 52 standard playing cards is used. The cards are dealt one at a time until the Queen of Spades (a unique card in the pack) is revealed. a What is the expected mean and standard deviation of the number of cards until the Queen of Spades is revealed? b In the game the player scores n 2 points if the Queen of Spades is the nth card revealed. Find the expected number of points scored.
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