Basics of ML & AL Week 2 Challenge

1)Calculate all 4 business moments using pen and paper for the below data set

1)  mean µ= ∑(x1) * p(x)

            = 0*0.35+1*0.25+ 2*0.15 + 3*0.15 + 4*0.10

            = 0 + 0.25 + 0.3 +0.45 +0.4

            = 1.4

2) Second business movement

Variance σ2 = ∑(x1-µx)^2 *P(x) 

               σ2 = (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

               σ2   = 0.686+0.04+0.054+0.384+0.676

               σ2   = 1.84

3) Third Business movement

 σ = 1.356

Skewness (SK) =∑((xi-µx)/σ)^3* P(x)

                    = ((0-1.4)/1.356)^3 *0.35 + ((1-1.4)/1.356)^3 *0.25 +  ((2-1.4)/1.356)^3 *0.15 + ((3-1.4)/1.356)^3                                *0.15 + ((4-1.4)/1.356)^3 *0.10

                   = -0.3851+ (-0.00641) + 0.0129 + 0.246 + 0.704

                   = 0.57139

4) Forth Business Movement

Kurtosis (Kt) =∑((xi-µx)/σ)^4 *p(x) -3

                   = (((0-1.4)/1.356)^4*0.35 +  ((1-1.4)/1.356)^4*0.25+ ((2-1.4)/1.356)^4*0.15+ ((3-                                                  1.4)/1.356)^4*0.15 + ((4-1.4)/1.356)^4*0.10)-3 

               = (0.3976 + 0.00189 + 0.00574 + 0.2907 + 1.3516)-3

              = -0.97665

              = -0.98

2)What is the significance of expected value when simple mean (Sum of all observations/number of observations) is already in place

 The application of simple mean and expected value are dependent on the nature of the case study. In smaller case studies where the random variable outcomes can be measured as the frequency of occurrences or each observation in the sample can be noted, the sample mean is calculated sum of all observations/number of observations`

However, in a case of an infinite number of observations, it is very difficult to note each and every outcome of the given dataset. In such cases, the probability of each outcome is determined using a sample study, and using this probability the expected value is determined using. The expected values reduce the analysis expenditure and serve the purpose of measuring the central tendency of the data set.

Mean or expected value, $\mu =\sum x.P\left(x\right)$

The expected value is an anticipated value for investment at some point in the future. in statistics and probability analysis, the expected value is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values. by calculating expected values, investors can choose the scenario most likely to give the desired outcome.

The sample mean varies with each sample. The difference in expected and sample mean differs in the application if you were going to calculate the mean from the frequency distribution i.e by multiplying the frequency with the variable and summing up and dividing by the total frequency we can get the sample mean. If we have the probability of occurrence of a random variable we can use the expectation formula. The expected value is calculated as follows for e.g the player plays occur as the probability of occurrence is given by the probability that they play zero days is 0.2 the probability that they play one day is 0.5 and the probability that they play two days is 0.3.

Here the days are random variables 0 days, 1 day, and 2 days the occurrence of appearance is 0.2,0.5, and 0.3.multiplying both will give 0,0.5 and 0.6, and summing up will give expected values as 1.1. The sample mean varies with the frequency of appearance in the occurrence of the variable hence the sample varies with the value.

Expected value and mean both are the same but we use these terms in different situations.Situation_1: suppose A, B, C, D, and their occurrences are given in this scenario simple mean itself is enough.But what if instead of a number of occurrences their probability is given.For example, in situation_2: the probability of A occurring is 0.50, the probability of B occurring is 0.29, for C is 0.21, and so on. In such a situation we calculate the mean of probability distribution and this is known as the expected value. Let us understand these terms with examples.

Mean is calculated by summing all observations divided by the number of observations. Mean is typically used when we want to calculate the average value of a given sample. This represents the average value of raw data that we have already collected. Expected value is used when we want to calculate the mean of a probability distribution. This represents the average value we expect to occur before collecting any data.

3)Having skewness in the curve considered to be bad in the analysis?

Skewness refers to a distortion or asymmetric 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.

Having skewness in the curve is not a bad analysis the skewness tells about whether the curve is symmetric or not and it also helps us to plot the different curves in the single one where the median and mode will be the same for different curves and thus tells us whether the curve is symmetric or not symmetric. Mean - Average of data sets, Median - Central value in the data set, Mode- Repeating number in the data set. 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.

A positive Skewed distribution is a 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 the mean is the greatest one followed by the median and then by mode. Well, The answer to that is the skewness of the distribution is on the right it cause 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.

A negative skewed distribution is a distribution with the tail on its left side. The value of skewness for a negative skewed distribution is less than zero. From the above, we can understand that skewness doesn`t have any bad effect on analysis rather it helps us identify certain characteristic of the data set just by looking at the graph. This proves to be very effective in data analytics and has no bad effects associated with it.

The general skewness is neither good nor bad without knowing the application of the data or the purpose for what it was obtained. Different applications have different meanings for skewed or non-skewed data. A distribution can have a positive skew no skew or a negative skew or even be undefined. 

4)Evaluate the probabilities for continuous normal distribution with given mean = 680 and standard deviation = 31

a)P(X<711)

b)P(X>740)

c)P(600<x<720)< p=""></x<720)<>

d)P(X=720)

me

z= X - Mean / Standard Deviation 

a) P(X<711)

  = 711- 680/31

  = 1

a) P(X<740) 

    = 740 - 680 /31

    = 1.935

c)P(600<x<720)< p=""></x<720)<>

1)    = 600 -680 /31

        = -2.58

 2)   =720-680 / 31 

       = 1.29

 P(-2.58 <Z< 1.29) = P(Z<1.29) - P(Z<-2.58)

                             = 0.9015 -0.0049

                             = 0.8966

                            = 89.66%

d)P(X=720)

 = 720 -680/ 31

 = 1.290

 find the z probability

P(z=1.29) = P(1.29<Z<Z1.29)

                = 0.9015-0.9015

                = 0

5) Explain the curve  on the right side

The curve shows in represents a normal distribution. The peak of the curve divides the curve and the area is symmetric about the y - axis with the mean or expected value located at the origin. the right side of the curve is divided using the standard deviation. The area under the curve between the mean and 1 standard deviation is 34.1 % of the total area. this means that 34.1% of the outcomes lie in this region.

 The distribution in the curve is perfect without any skewness. 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 probability for value further away from the mean taper off equally in both directions.

first standard deviation from the mean is approximately 34.1% to wither the left or the right side of the mean. This means that the standard deviation of a normal distribution can lie anywhere within the space of + 34.1 or -34.1 % So approximately 68.2% of the values in the distribution fall within first standard deviation of the mean. Similiarly beyond the first standard deviation 13.6% to the left or to the right of the first standard deviation represent the values that fall within the second standard deviation. This means to say that 95.4% of the values in the distribution fall second standard deviation of the mean. This continues to the third standard deviation and so on with area under the curve increasing`

The standard deviation determines the width of the curve and it tightens or expands the width of the distribution along the x- axis.smaller standard deviation indicates the data clustered tightly around the mean and normal distribution will be taller and vice versa. We can see in above bell curve percentage shows the area covered under the curve and divisions along the x-axis shows standard deviation and central line is mean.