Normal Distribution

� Properties of a random variable

1 . � : Mean 2 . � : Standard deviation

2

X ~ N ( �� , ) following normal distribution :

3 . The mean , the median and the mode are the same

4 . The normal curve representing the distribution is a bell-shaped curve which is symmetric about the middle vertical line 5 . P ( X � �) � P ( X � �) � 0.5 6 . The total area under the curve is 1

Linear Combinations of Variables

� Properties of linear transformations of independent random variables : 1 . E ( aX �b) � aE ( X ) � b 2 .

2

Var ( aX �b) � a Var ( X ) 3 . E ( a1 X1 � a2 X

2

� � a X ) � a1E ( X1) � a2E ( X

2) � � a E ( X )

4 .

n n n n

Var ( a X � a X � � a X ) � a Var ( X ) � a Var ( X ) � � a Var ( X )

Point Estimation

� Central limit theorem : 1 .

X N ��

2 i

~ ( , ) X1� X

2

� � X n

2 . X �

: Sample mean n

2

� � � 3 . X ~ N��,

� when n is sufficiently large � n � www . seprodstore . com

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