challenge 2 by vishal gautam
1. (i) First Business Moment : Measure of tendency
Mean(`mu`) = `Sigma (x).P(X) =`Expected value`
`mu`= (0).(0.35)+(1).(0.25)+(2).(0.15)+(3).(0.15)+(4).(0.10)
= 0+0.25+0.30+0.45+0.40
=1.4
(ii) Second Business Moment : Measure of dispersion
Variance (`sigma^2`) = `Sigma(x-mu)^2.(P(X))`
=(0-1.4)^2.(0.35)+(1-1.4)^2.(0.25)+(2-1.4)^2.(0.15)+(3-1.4)^2.(0.15)+(4- 1.4)^2.(0.10)
=0.686+0.04+0.054+0.384+0.676
=1.84
(iii) Third Business Moment : Skewness
Sk = `Sigma ((x-mu)/sigma)^3.(P(X))`
= -0.3851-0.006417+0.01299+0.2464+0.70492
= 0.572793
(iv) Fourth Business Moment : Kurtosis
Kt = `Sigma ((x-mu)/sigma)^4.(P(X))`
= 0.3976+0.001892+0.005749+0.2907+1.3516
= 2.047541
2. The significance of expected value when simple mean is already in place is that when there is uniform probability distribution , we can directly use simple mean,otherwise we use expected mean for better analytical results and predictions.
In a more detailed way,we can say that,
- Expected value is used in case of random variables.Since, the average is defined as the sum of all the elements divided by sum of their frequencies.But for the case of probability distribution we cannot describe a random variable in terms of its frequency beforehand,thus we use the probability instead.Conceptually,probability of an element is frequency of an event divided by size of sample space.Thus ,the average in case of random variable can be given by sum of probabilities multiplied by its respective event.
- Average on the otherhand is used in case where we have the knowledge of frequencies of individual elements and totoal counts of elements,for example : in case of known data set or sample we can simply use the fundamental definition of the average to calculate it.
3. A positive skew could be good or bad , depending on the mean. A positive mean with a positive skew is good, while a negative mean with a positive skew is not good. If a dataset has a positive skew , but the mean of the returns is negative, it means that overall performance is negative, but the outlier months are positive.A negative skew is generally not good , because it highlights the risk of left tail events or what are sometimes referred to as \"Black swan events\".While a consistent and steady track record with a positive mean would be a great thing, if the track record has a negative skew then you should proceed with caution.
4. Given data,
Mean `mu` = 680
Standard deviation `sigma` = 31
formula for z - score = `(x-mu)/sigma`
(a) now probability for P(x>711) = 1-P(x<=711)
z = (711-680)/31=1
now by using z tables,we get
P(x<=711) = 0.8413
P(x>711) = 1-0.8413=0.1587
(b) P(x>740)
z = (740-680)/31=1.9354
P(x>740)=1-P(x<=740)
now by using z tables,we get
P(x<=740) = 0.9732
P(x>740) = 1-0.9732=0.0268
(c) P(600<x<720)
P(600<x<720) = P(x<720)-P(x<600)
z(600)=(600-680)/31=-2.58
z(720)= (720-680)/31=1.29
now by using z tables,we get
P(x<720) = 0.9015 and P(x<600) = 0.0049
P(600<x<720) = 0.9015-0.0049 = 0.8966
(d) P(x=720) = 1-P(x<720)-P(x>720)
now by using z tables,we get
P(x<720)=0.90153
P(x>720)=0.09847
P(x=720) = 1-0.9015-0.09847=0
5. It shows us that what percentage of your data falls within a certain number of standard deviations from the mean:
- 68% of the data falls within one standard deviation of the mean.
- 95% of the data falls within two standard deviations of the mean.
- 99.7% of the data falls within three standard deviations of the mean.
The standard deviation controls the spread of the distribution. A smaller standard deviation indicates that the data is tightly clustered around the mean;the normal distribution will be taller. A larger standard deviation indicates that the data is spread out the mean ; the normal distribution will be flatter and wider.
It also indicates properties of a normal distribution curve which are :
- The mean,mode and median are all equal.
- The curve is symmetric at the centre.
- Exactly half of the values are to the left of centre and exactly half of the values are to the right.
-The total area under the curve is 1.