3. Continuous Random Variables
3. Continuous Random Variables
3. Continuous Random Variables
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∫<br />
∫<br />
(b) Mean = = 9491 hours.<br />
Standard deviation =√ = √<br />
Normal distribution<br />
[ ]<br />
[ ]<br />
= 9491 hours.<br />
Statistics and probability: 3-5<br />
The continuous random variable has the Normal distribution if the pdf is:<br />
√<br />
The parameter � is the mean and and the<br />
variance is � 2 . The distribution is also<br />
sometimes called a Gaussian distribution.<br />
The pdf is symmetric about �. X lies between �<br />
- 1.96� and � + 1.96� with probability 0.95 i.e.<br />
X lies within 2 standard deviations of the mean<br />
approximately 95% of the time.<br />
Normalization<br />
[non-examinable]<br />
cannot be integrated analytically for general ranges, but the full range can be<br />
integated as follows. Define<br />
I �<br />
�<br />
�<br />
��<br />
dx e<br />
2<br />
2<br />
�(<br />
x��<br />
)<br />
�x<br />
2 �<br />
2<br />
2�<br />
2�<br />
�<br />
�<br />
��<br />
dx e<br />
Then switching to polar co-ordinates we have
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