Basics of ML & AL Week 2 Challenge

Ans no 1:- photo

First business moment : Mean =	1.4
2.Second business moment : Variance=	1.84
Third business moment : sk=	0.572138895
Fourth business moment : kt=	(2.04489603024575)-3	-0.95510397

Ans no 2 :- Arithmetic mean is simple average of numbers not weighted by anything. However in Expected Value, the numbers are weighted by their probability.
1.When we consider the occourances of quantities to be same in experiment of calculating aritheatic mean and expected value then arithametic mean will give us same answer as the expected value .

Arithmetic mean : is the total of the sum of all values in a collection of numbers divided by the number of numbers in a collection. It is calculated in the following way:
x1+x2+x3+x4+......n">x1+x2+x3+x4+......nx1+x2+x3+x4+......Xn

x1,x2,x3.....=outcomes
n=number of outcomes

Ans no 3 :- Skewness : Skewness refers to a distortion or asymmetry that deviates from the symmetrical bell curve, or normal distribution in a set of data. If the curve is shifted to the left or to the right, it is said to be skewed. Skewness can be quantified as a representation of the extent to which a given distribution varies from a normal distribution. A normal distribution has a skew of zero

Why is skewness important ?

Linear models work on the assumption that the distribution of the independent variable and the target variable are similar. Therefore, knowing about the skewness of data helps us in creating better linear models.

Let’s take a look at the below distribution. It is the distribution of horsepower of cars:

You can clearly see that the above distribution is positively skewed. Now, let’s say you want to use this as a feature for the model which will predict the mpg (miles per gallon) of a car.

Since our data is positively skewed here, it means that it has a higher number of data points having low values, i.e., cars with less horsepower. So when we train our model on this data, it will perform better at predicting the mpg of cars with lower horsepower as compared to those with higher horsepower.

You can see that our distribution is positively skewed and most of the outliers are present on the right side of the distribution.

Positively skewed :

A positively skewed distribution is the distribution with the tail on its right side. The value of skewness for a positively skewed distribution is greater than zero. As you might have already understood by looking at the figure, the value of mean is the greatest one followed by median and then by mode.

Well, the answer to that is that the skewness of the distribution is on the right; it causes the mean to be greater than the median and eventually move to the right.
Also, the mode occurs at the highest frequency of the distribution which is on the left side of the median. Therefore, mode < median < mean.

Negatively skewed :

A negatively skewed distribution is the distribution with the tail on its left side. The value of skewness for a negatively skewed distribution is less than zero.
You can also see in the above figure that the mean < median < mode.

From the above we can understand that skewness doesn't have any bad effect on analysis rather it helps us identify certain characteristics of the data set just by looking at the graph .

Ans no 4:-

So , Z =( X-μσ">X−μ)/σ

where ,μ=mean">μ=mean= 680
            σ=">σ=Standard deviation = 31

a)P(X<711)

Z = (X-μσ">X−μ)/σ

   = (711 - 680)/31

   = 1

Since we converted X to Z and also 711 to 1 ,(X<711) converts to (Z<1)
From the Z table at Z=0
 probablity is P(Z<1) = 0.8413

b)Converting P(X>740) into Z score ,

Z = (X-μσ">X−μ)/σ

   = (740 - 680)/31

   = 1.94

Since we converted X to Z and also 711 to 1.94 ,(X>711) converts to (Z>1.94)
From the Z table at Z=0
So , probablity of Z<1.94 is P(Z<1.94) = 0.9738

But the probablity of Z>1.94 is P(Z>1.94) is (1-0.9738) = 0.0262

P(Z>1.94) = 0.0262

c)Converting P(600

Z = (X-μσ">X−μ)/σ

For X = 600 ,

Z = (600 - 680)/31

   = -2.5805 ~ -2.57

Since we converted X to Z and also 600 to -2.57 ,(X>600) converts to (Z>-2.57)
From the Z table at Z=0
So , probability of Z>-2.57 is P(Z>-2.57) = 0.0051

Z = (X-μσ">X−μ)/σ

For X = 720 ,

Z = (720 - 680)/31

   = 1.2903226 ~ 1.28

Since we converted X to Z and also 720 to 1.28 ,(X<720) converts to (Z<1.28)
From the Z table at Z=0
So , probability of Z<1.28 is P(Z<1.28) = 0.8997

The actual total probablity of P(600
which is 0.8997 - 0.0051 = 0.8946


d)P(X=720)
This is an interesting case of finding the probablity of an occourance within the range of 720 and 720 ,in other words would mean P(720
Z = (X-μσ">X−μ)/σ

For X = 720 ,

Z = (720 - 680)/31

   = 1.2903226 ~ 1.29

So , probability of Z=1.29 is P(Z=1.29) = 0.8997

So,
P(1.3= 0.8997-0.8997
=0

Ans no 5 :-

The  curve is the normal distribution curve for the random variable and the percentage is the standard deviations from the mean.The normal distribution is a probability function that describes how the values of a variable are distributed. It is a symmetric distribution where most of the observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions.

1σ (std.Deviation) from the mean is arounnd 34.1 either side of the mean and hence the values lies between -34.1 and +34.1,so 68.2% of values falls in the 1 std.deviation.

 2σ (std.Deviation) from the mean is taken as 34.1+34.1+13.6+13.6=95.4% approximately the values falls in the 2 std.deviation from the mean 

Likewise 3σ (std.Deviation) rom the mean is taken as 34.1+34.1+13.6+13.6+2.1+2.1=99.7% approximately the values falls in the 3 std.deviation from the mean 

The perfect normal distribution curve  will have no skewness and distribution follows the symmetry.

Formula to find z scores Z=x-μσ">Z=(x−μ)/σ">σ
σ">σ= Standard deviation
μ">μ= Mean
x=">x=Value of x(value being examined)

a) The SND (i.e. z-distribution) is always the same shape as the raw score distribution. For example, if the distribution of raw scores if normally distributed, so is the distribution of z-scores.

b) The mean of any SND always = 0.

c) The standard deviation of any SND always = 1. Therefore, one standard deviation of the raw score (whatever raw value this is) converts into 1 z-score unit.

1. The SND allows researchers to calculate the probability of randomly obtaining a score from the distribution (i.e. sample).
For example, there is a 68% probability of randomly selecting a score between -1 and +1 standard deviations from the mean .
The probability of randomly selecting a score between -1.96 and +1.96 standard deviations from the mean is 95%