variance and standard deviation of a discrete random variable
TRANSCRIPT
![Page 1: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/1.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.X P(x)0 0.21 0.32 0.23 0.24 0.1
![Page 2: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/2.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.X P(x)0 0.21 0.32 0.23 0.24 0.1
π 2=β [π₯2 βπ (π₯ )]βπ2
![Page 3: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/3.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.X P(x)0 0.21 0.32 0.23 0.24 0.1
π 2=β [π₯2 βπ (π₯ )]βπ2
First find the mean.
![Page 4: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/4.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.X P(x)0 0.21 0.32 0.23 0.24 0.1
π 2=β [π₯2 βπ (π₯ )]βπ2
First find the mean.
π=β [π₯ βπ (π₯ )]
![Page 5: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/5.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.X P(x)0 0.21 0.32 0.23 0.24 0.1
π=β [π₯ βπ (π₯ )]Create a column of xβP(x)
π 2=β [π₯2 βπ (π₯ )]βπ2
![Page 6: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/6.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x)0 0.21 0.32 0.23 0.24 0.1
π=β [π₯ βπ (π₯ )]Create a column of xβP(x)
π 2=β [π₯2 βπ (π₯ )]βπ2
![Page 7: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/7.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x)0 0.2 0(0.2) = 01 0.32 0.23 0.24 0.1
π=β [π₯ βπ (π₯ )]Create a column of xβP(x)
π 2=β [π₯2 βπ (π₯ )]βπ2
![Page 8: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/8.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x)0 0.2 0(0.2) = 01 0.3 1(0.3) = 0.32 0.23 0.24 0.1
π=β [π₯ βπ (π₯ )]Create a column of xβP(x)
π 2=β [π₯2 βπ (π₯ )]βπ2
![Page 9: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/9.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x)0 0.2 0(0.2) = 01 0.3 1(0.3) = 0.32 0.2 2(0.2) = 0.43 0.24 0.1
π=β [π₯ βπ (π₯ )]Create a column of xβP(x)
π 2=β [π₯2 βπ (π₯ )]βπ2
![Page 10: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/10.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x)0 0.2 0(0.2) = 01 0.3 1(0.3) = 0.32 0.2 2(0.2) = 0.43 0.2 3(0.2) = 0.64 0.1
π=β [π₯ βπ (π₯ )]Create a column of xβP(x)
π 2=β [π₯2 βπ (π₯ )]βπ2
![Page 11: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/11.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x)0 0.2 0(0.2) = 01 0.3 1(0.3) = 0.32 0.2 2(0.2) = 0.43 0.2 3(0.2) = 0.64 0.1 4(0.1) = 0.4
π=β [π₯ βπ (π₯ )]Create a column of xβP(x)
π 2=β [π₯2 βπ (π₯ )]βπ2
![Page 12: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/12.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x)0 0.2 0(0.2) = 01 0.3 1(0.3) = 0.32 0.2 2(0.2) = 0.43 0.2 3(0.2) = 0.64 0.1 4(0.1) = 0.4
π=β [π₯ βπ (π₯ )]
Sum the column of xβP(x)
π 2=β [π₯2 βπ (π₯ )]βπ2
![Page 13: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/13.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x)0 0.2 0(0.2) = 01 0.3 1(0.3) = 0.32 0.2 2(0.2) = 0.43 0.2 3(0.2) = 0.64 0.1 4(0.1) = 0.4
π=β [π₯ βπ (π₯ )]
Sum the column of xβP(x)
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
![Page 14: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/14.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x)0 0.2 0(0.2) = 01 0.3 1(0.3) = 0.32 0.2 2(0.2) = 0.43 0.2 3(0.2) = 0.64 0.1 4(0.1) = 0.4
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
![Page 15: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/15.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x)0 0.2 0(0.2) = 01 0.3 1(0.3) = 0.32 0.2 2(0.2) = 0.43 0.2 3(0.2) = 0.64 0.1 4(0.1) = 0.4
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2
![Page 16: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/16.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2
0 0.2 0(0.2) = 01 0.3 1(0.3) = 0.32 0.2 2(0.2) = 0.43 0.2 3(0.2) = 0.64 0.1 4(0.1) = 0.4
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2
![Page 17: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/17.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2
0 0.2 0(0.2) = 0 02 = 01 0.3 1(0.3) = 0.32 0.2 2(0.2) = 0.43 0.2 3(0.2) = 0.64 0.1 4(0.1) = 0.4
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2
![Page 18: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/18.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2
0 0.2 0(0.2) = 0 02 = 01 0.3 1(0.3) = 0.3 12 = 12 0.2 2(0.2) = 0.43 0.2 3(0.2) = 0.64 0.1 4(0.1) = 0.4
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2
![Page 19: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/19.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2
0 0.2 0(0.2) = 0 02 = 01 0.3 1(0.3) = 0.3 12 = 12 0.2 2(0.2) = 0.4 22 = 43 0.2 3(0.2) = 0.64 0.1 4(0.1) = 0.4
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2
![Page 20: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/20.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2
0 0.2 0(0.2) = 0 02 = 01 0.3 1(0.3) = 0.3 12 = 12 0.2 2(0.2) = 0.4 22 = 43 0.2 3(0.2) = 0.6 32 = 94 0.1 4(0.1) = 0.4
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2
![Page 21: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/21.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2
0 0.2 0(0.2) = 0 02 = 01 0.3 1(0.3) = 0.3 12 = 12 0.2 2(0.2) = 0.4 22 = 43 0.2 3(0.2) = 0.6 32 = 94 0.1 4(0.1) = 0.4 42 = 16
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2
![Page 22: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/22.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2
0 0.2 0(0.2) = 0 02 = 01 0.3 1(0.3) = 0.3 12 = 12 0.2 2(0.2) = 0.4 22 = 43 0.2 3(0.2) = 0.6 32 = 94 0.1 4(0.1) = 0.4 42 = 16
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2βP(x)
![Page 23: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/23.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2 x2βP(x)0 0.2 0(0.2) = 0 02 = 01 0.3 1(0.3) = 0.3 12 = 12 0.2 2(0.2) = 0.4 22 = 43 0.2 3(0.2) = 0.6 32 = 94 0.1 4(0.1) = 0.4 42 = 16
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2βP(x)
![Page 24: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/24.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2 x2βP(x)0 0.2 0(0.2) = 0 02 = 0 0(0.2) = 01 0.3 1(0.3) = 0.3 12 = 12 0.2 2(0.2) = 0.4 22 = 43 0.2 3(0.2) = 0.6 32 = 94 0.1 4(0.1) = 0.4 42 = 16
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2βP(x)
![Page 25: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/25.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2 x2βP(x)0 0.2 0(0.2) = 0 02 = 0 0(0.2) = 01 0.3 1(0.3) = 0.3 12 = 1 1(0.3) = 0.32 0.2 2(0.2) = 0.4 22 = 43 0.2 3(0.2) = 0.6 32 = 94 0.1 4(0.1) = 0.4 42 = 16
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2βP(x)
![Page 26: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/26.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2 x2βP(x)0 0.2 0(0.2) = 0 02 = 0 0(0.2) = 01 0.3 1(0.3) = 0.3 12 = 1 1(0.3) = 0.32 0.2 2(0.2) = 0.4 22 = 4 4(0.2) = 0.83 0.2 3(0.2) = 0.6 32 = 94 0.1 4(0.1) = 0.4 42 = 16
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2βP(x)
![Page 27: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/27.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2 x2βP(x)0 0.2 0(0.2) = 0 02 = 0 0(0.2) = 01 0.3 1(0.3) = 0.3 12 = 1 1(0.3) = 0.32 0.2 2(0.2) = 0.4 22 = 4 4(0.2) = 0.83 0.2 3(0.2) = 0.6 32 = 9 9(0.2) = 1.84 0.1 4(0.1) = 0.4 42 = 16
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2βP(x)
![Page 28: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/28.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2 x2βP(x)0 0.2 0(0.2) = 0 02 = 0 0(0.2) = 01 0.3 1(0.3) = 0.3 12 = 1 1(0.3) = 0.32 0.2 2(0.2) = 0.4 22 = 4 4(0.2) = 0.83 0.2 3(0.2) = 0.6 32 = 9 9(0.2) = 1.84 0.1 4(0.1) = 0.4 42 = 16 16(0.1) = 1.6
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Create a column of x2βP(x)
![Page 29: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/29.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2 x2βP(x)0 0.2 0(0.2) = 0 02 = 0 0(0.2) = 01 0.3 1(0.3) = 0.3 12 = 1 1(0.3) = 0.32 0.2 2(0.2) = 0.4 22 = 4 4(0.2) = 0.83 0.2 3(0.2) = 0.6 32 = 9 9(0.2) = 1.84 0.1 4(0.1) = 0.4 42 = 16 16(0.1) = 1.6
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Sum the column of x2βP(x)
![Page 30: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/30.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2 x2βP(x)0 0.2 0(0.2) = 0 02 = 0 0(0.2) = 01 0.3 1(0.3) = 0.3 12 = 1 1(0.3) = 0.32 0.2 2(0.2) = 0.4 22 = 4 4(0.2) = 0.83 0.2 3(0.2) = 0.6 32 = 9 9(0.2) = 1.84 0.1 4(0.1) = 0.4 42 = 16 16(0.1) = 1.6
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
π 2=β [π₯2 βπ (π₯ )]βπ2
Sum the column of x2βP(x)Ξ£[x2βP(x)]=4.5
![Page 31: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/31.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2 x2βP(x)0 0.2 0(0.2) = 0 02 = 0 0(0.2) = 01 0.3 1(0.3) = 0.3 12 = 1 1(0.3) = 0.32 0.2 2(0.2) = 0.4 22 = 4 4(0.2) = 0.83 0.2 3(0.2) = 0.6 32 = 9 9(0.2) = 1.84 0.1 4(0.1) = 0.4 42 = 16 16(0.1) = 1.6
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
4.5 β 1.72
1.61
Ξ£[x2βP(x)]=4.5
Variance
![Page 32: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/32.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2 x2βP(x)0 0.2 0(0.2) = 0 02 = 0 0(0.2) = 01 0.3 1(0.3) = 0.3 12 = 1 1(0.3) = 0.32 0.2 2(0.2) = 0.4 22 = 4 4(0.2) = 0.83 0.2 3(0.2) = 0.6 32 = 9 9(0.2) = 1.84 0.1 4(0.1) = 0.4 42 = 16 16(0.1) = 1.6
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
4.5 β 1.72
1.61
Ξ£[x2βP(x)]=4.5
Variance
π=βπ2
![Page 33: Variance and standard deviation of a discrete random variable](https://reader036.vdocuments.us/reader036/viewer/2022062412/586f8f451a28ab54768b74d3/html5/thumbnails/33.jpg)
Example 7: Find the variance and standard deviation of the probability distribution.
X P(x) xβP(x) x2 x2βP(x)0 0.2 0(0.2) = 0 02 = 0 0(0.2) = 01 0.3 1(0.3) = 0.3 12 = 1 1(0.3) = 0.32 0.2 2(0.2) = 0.4 22 = 4 4(0.2) = 0.83 0.2 3(0.2) = 0.6 32 = 9 9(0.2) = 1.84 0.1 4(0.1) = 0.4 42 = 16 16(0.1) = 1.6
π=β [π₯ βπ (π₯ )]=1.7
Ξ£[xβP(x)]=1.7
4.5 β 1.72
1.61
Ξ£[x2βP(x)]=4.5
Variance
π=βπ2=β1.61β1.27
Standard Deviation