![[Maple OLE 2.0 Object]](images/binostat1.gif)
Guess what the short cuts are
In this section we will go through several examples the long way using a computer. As we do this, see if you note any patterns that suggest a short cut. For each example note what n, p, q, the mean, and the variance are. Since the standard deviation is the square root of the variance, do not bother to note the value of the standard deviation. After you've seen several examples, you may be able to guess the short cut.
Example 1
Compute the mean, variance and standard deviation for the binomial distribution with n = 4 and p = .25.
Solution .
In addition to the probability of each x, the tables below show the computations of the mean, variance, and standard deviation.
x prob(x) x*prob(x) ((x - mean)^2)*prob(x)
---------------------------------------------------------------
0 .31640625 0 .3164062500
1. .42187500 .42187500 0
2. .21093750 .42187500 .2109375000
3. .4687500E-1 .14062500 .1875000000
4. .390625E-2 .1562500E-1 .3515625000E-1
SUM 1.00000000 1.00000000 .7500000000
mean variance
.8660254038
standard deviation
Example 2
Now use n = 4 and p = .5.
Solution .
x prob(x) x*prob(x) ((x - mean)^2)*prob(x)
---------------------------------------------------------------
0 .625E-1 0 .2500000000
1. .2500 .2500 .2500000000
2. .3750 .7500 0
3. .2500 .7500 .2500000000
4. .625E-1 .2500 .2500000000
SUM 1.0000 2.0000 1.000000000
mean variance
1.000000000
standard deviation
Example 3
Use n = 5 and p = .3.
Solution .
x prob(x) x*prob(x) ((x - mean)^2)*prob(x)
---------------------------------------------------------------
0 .16807 0 .3781575000
1. .36015 .36015 .9003750000E-1
2. .30870 .61740 .7717500000E-1
3. .13230 .39690 .2976750000
4. .2835E-1 .11340 .1771875000
5. .243E-2 .1215E-1 .2976750000E-1
SUM 1.00000 1.50000 1.050000000
mean variance
1.024695077
standard deviation
Example 4
Use n = 8 and p = .25.
Solution .
x prob(x) x*prob(x) ((x - mean)^2)*prob(x)
----------------------------------------------------------------
0 .1001129150 0 .4004516600
1. .2669677734 .2669677734 .2669677734
2. .3114624023 .6229248046 0
3. .2076416016 .6229248048 .2076416016
4. .8651733398E-1 .3460693359 .3460693359
5. .2307128906E-1 .1153564453 .2076416015
6. .3845214844E-2 .2307128906E-1 .6152343750E-1
7. .3662109375E-3 .2563476563E-2 .9155273438E-2
8. .1525878906E-4 .1220703125E-3 .5493164062E-3
SUM .9999999999 2.000000000 1.500000000
mean variance
1.224744871
standard deviation
Example 5
Use n = 6 and p = .3.
Solution .
x prob(x) x*prob(x) ((x - mean)^2)*prob(x)
---------------------------------------------------------------
0 .117649 0 .3811827600
1. .302526 .302526 .1936166400
2. .324135 .648270 .1296540000E-1
3. .185220 .555660 .2667168000
4. .59535E-1 .238140 .2881494000
5. .10206E-1 .51030E-1 .1045094400
6. .729E-3 .4374E-2 .1285956000E-1
SUM 1.000000 1.800000 1.260000000
mean variance
1.122497216
standard deviation
The following table summarizes the results of these examples
Do you have an conjectures about short cuts that give the mean and variance?