Analysis and Approaches for IBDP Maths Ebook 2 | Page 251

79
Paper 1 Section A – Problems Involving Statistics
Example
The random variable X has the probability density function
π π
f ( x) � sin x , 0�x
� 4.
8 4
Find
(a) E( X ) ;
(b)
the interquartile range.
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[5]
Solution
(a)
4 π π
E( X ) �� x�
sin xdx
(A1) for substitution
0
8 4
�
Let � � cos x . 4
19
d � 1
� � � sin � x � � d�
�
� sin � xdx
dx
4 4 2 8 4
4 1 � � �
�E( X ) � � � xd cos x
0
� �
2 � 4 �
4
1 �
4 � 1
2 4 � 0 �
0
4 2
� � � �
E( X ) �
�
� xcos x �� cos xd
� x � A1
� � � �
4
1 � � 4 1
2 4 �
�
0
0
2 4
�
�
E( X ) �
�
� xcos x � cos xdx
� �
4 4
1 � 2 �
E( X ) � � �
� xcos x � � sin x
�
2 4 �
�
��
4 �
� � � �
E( X ) � (2 �0) � (0 � 0)
0 0
A1
E( X ) � 2
A1
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