In: Statistics and Probability
X f(X)
0 |
0.10 |
1 |
0.15 |
2 |
0.30 |
3 |
0.20 |
4 |
0.15 |
5 |
0.10 |
The given PDF is graphed as a bar chart as:
The following table would be used for the computations now:
x | f(x) | xf(x) | x^2*f(x) |
0 | 0.1 | 0 | 0 |
1 | 0.15 | 0.15 | 0.15 |
2 | 0.3 | 0.6 | 1.2 |
3 | 0.2 | 0.6 | 1.8 |
4 | 0.15 | 0.6 | 2.4 |
5 | 0.1 | 0.5 | 2.5 |
1 | 2.45 | 8.05 |
The mean and standard deviation are computed using the above table last row of sum of columns as:
Therefore 2.45 is the required mean value here.
The standard deviation now is computed here as:
Therefore 1.43 is the required standard deviation here.
These are plotted on the graph as well as the ranges are plotted here as:
Mean - Std Dev = 2.45 - 1.43 = 1.02
Mean + Std Dev = 2.45 + 1.43 = 3.88
Mean - 2*Std Dev = 2.45 - 2*1.43 = -0.41
Mean + 2*Std Dev = 2.45 + 2*1.43 = 5.31
Note that the middle line represents the mean while the other two lines represent distances 1 standard deviation from mean from either side.
Note that P(X = 2) + P(X = 3) lies within 1 standard deviation of the mean that is 0.3 + 0.2 = 0.5 that is 50%
Also, we can see that within 2 standard deviations of the mean which is from (-0.41, 5.31), 100% of the observations lies within this range.
According to the empirical rule, 95% of the observations lies within 2 standard deviations of the mean and 68% lies within 1 standard deviation of the mean. Therefore empirical rule is not valid here ( anyways it is valid for only normal distribution )
According to chebyshev's inequality theorem, at least 1 - 1/k2 of the observations lies within k standard deviations of the mean.
For k = 1, 1 - 1/k2 = 0, therefore at least 0% lies
within 1 standard deviation. As 50% lies this is applicable
here.
Also for k = 2, 1 - 1/22 = 75%, as 100% lies therefore
at least 75% is still valid. Therefore chebyshev's theorem
is applicable for the given probability distribution
here.