The Binomial Distribution: Mean and Variance © Christine Crisp “Teach A Level Maths” Statistics...
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The Binomial The Binomial Distribution: Mean Distribution: Mean and Variance and Variance © Christine Crisp “ “ Teach A Level Teach A Level Maths” Maths” Statistics 1 Statistics 1
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Transcript of The Binomial Distribution: Mean and Variance © Christine Crisp “Teach A Level Maths” Statistics...
The Binomial The Binomial Distribution: Mean Distribution: Mean
and Varianceand Variance
© Christine Crisp
““Teach A Level Teach A Level Maths”Maths”
Statistics 1Statistics 1
The Binomial Distribution – Mean and Variance
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
Statistics 1
AQAMEI/OCR
OCR
The Binomial Distribution – Mean and Variance
There are very simple formulae that we can use to find the mean and variance of a Binomial Distribution.I’ll illustrate the formulae with an example.
The Binomial Distribution – Mean and Variance
Suppose .)40,3(~ BX The probability distribution is )0(XP
)1(XP )2(XP )3(XP
2160432028800640
3210x
2160 4320 2880 0640)( xXP
Can you find the mean using the formula for any discrete probability distribution?
mean, =
)( xXxP 21600 ...43201 21
If you’ve forgotten how to calculate the mean, the results written in a table may remind you:
Can you spot the link to the parameters of the distribution ( n = 3 and p = 0·4 )?
The Binomial Distribution – Mean and Variance
mean, =
)0(XP )1(XP )2(XP )3(XP
2160432028800640
3210x
2160 4320 2880 0640)( xXP
Can you find the mean using the formula for any discrete probability distribution?
)( xXxP 21600 ...43201 21
If you’ve forgotten how to calculate the mean, the results written in a table may remind you:
ANS: so,40321 np
Suppose .)40,3(~ BX The probability distribution is
The Binomial Distribution – Mean and Variance
3210x
2160 4320 2880 0640)( xXP
22 )( xXPx
216002 432012
The formula for the variance of any discrete probability distribution is
variance npq
Now for the variance:
288022 064032 221720
The link with the parameters isn’t so easy to spot this time but it’s easy to remember. It is
)40,3(~ BX
9410x2
The Binomial Distribution – Mean and Variance
Solution:
variance npq
33010
e.g.1 Find the mean and variance of the random variable X where
)30,10(~ BX
npmean,
SUMMARY
The mean of a Binomial Distribution is given by
npmean,
The variance is given byvariance npq
12703010 variance
The Binomial Distribution – Mean and Variance
Exercise
Solutions:
1. Find the mean and variance of the random variable X where
)50,20(~ BX(a)
)020,7(~ BX
(b)
(c)
(d)
)30,15(~ BX
)250,12(~ BX
(a)
50,50,20 qpn mean 10npvariance 5npq
(b)
70,30,15 qpn mean 54npvariance 153npq
The Binomial Distribution – Mean and Variance
Exercise
Solutions:
1. Find the mean and variance of the random variable X where
)50,20(~ BX(a)
)020,7(~ BX
(b)
(c)
(d)
)30,15(~ BX
)250,12(~ BX
(c)
980,020,7 qpn mean 140npvariance 13720npq
(d)
750,250,12 qpn mean 3npvariance 252npq
The following slide contains repeats of information on earlier slides, shown without colour, so that it can be printed and photocopied.
The Binomial Distribution – Mean and Variance
variance npq
SUMMARY
npmean,
The variance is given by
variance npq
The mean of a Binomial Distribution is given by
Solution: 33010
e.g.1 Find the mean and variance of the random variable X where
)30,10(~ BX
12703010 variance
npmean,
