Basics of ML & AL Week 2 Challenge

Q1.

1.First Business Moment : measure of Central Tendency :

Mean(µ) = Σ(xi).P(x) = 1.4 0⋅0.35+1⋅0.25+2⋅0.15+3⋅0.15+4⋅0.1">

2.Second Business Moment : measure of variance(σ^2)  =  Σ(xi-µ)^2.P(x)

variance(σ^2)  =  Σ(xi-µ)^2.P(x) = 1.84

3.Thid Business Moment : Skewness(sk) : The third business moment is the skewness. Skewness is a measure of the symmetry in distribution. The skewness for a normal distribution is zero. A positive skewness indicates that the size of the right-handed tail is larger than the left-handed tail and vice-versa

sk=  Σ((xi - µ)/σ)^3.P(x) = 0.57

4.Fourth Business Moment :Kurtosis(kt) :The fourth business moment is the kurtosis Kurtosis is a measure of the combined sizes of the two tails.  It measures the amount of probability in the tails.  The value is often compared to the kurtosis of the normal distribution, which is equal to 3. 

Kt = Σ((xi - µ)/σ)^4.P(x) - 3

Kt  = 2.04 - 3

Kt = -0.96 

Q2.

The applicablity 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 frequency of occurences or each observation in the

sample can be noted, the simple mean is calculated as 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 data set. In such cases, the probability of each outcome is determined using a sample study and using this

probability the expected value is determined using: Mean(µ) = Σ(xi).P(x)

The expected values reduces the analysis expenditure and serves the purpose of the measuring the central tendency of

the data set.

Q3.

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.

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.

mode < median>

Negative 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.

mean < median>

Q4.  Given Data:

 mean = 680 and standard deviation = 31.

1).P(X<771>

P(X<771>

p(µ + 1σ)

in the normal distribution the area under the curve on the leaft ,which is the probability of the outcomes , is 0.5

Also area under the curve from origin to 1σ is 0.34

p(X<771>

B)P(X > 740):

Value of z corresponding to X = 740,

Z = X - µ / σ

Z = 740 - 680/ 31 = 1.935= 1.94

Using the Z-table ,probability  p(Z< 1>

Now, p(X>740) = 1 - p(X<740>

C) P(600 < x>

z = (X-µ) /σ

Z = (600-680)/31

Z = -2.5805 ~ - 2.57

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

X= 720

z = (X-µ) /σ

Z = (720-680)/31

   = 1.2903226 ~ 1.28

We converted X to Z and also 720 to1.28, (X< 720 xss=removed>

So Probabilty of P(600720) is P(-2.571.28) 

which is 0.8997 - 0.0051 = 0.8946.

P(600

d). P(x = 720)

The is an intersting case of finding the probability of an occourance within the range of 720 and 720 ,in other words would mean p(720

X = 720

z = (X-µ) /σ

  =(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

Z =0

Q5.

The above image show z scores on the horizontal axis and the probablity distribution on the vertical axis .

1.The value of the z-score tells you how many standard deviations you are away from the mean. If a z-score is equal to 0, it is on the mean.

2.A positive z-score indicates the raw score is higher than the mean average. For example, if a z-score is equal to +1, it is 1 standard deviation above the mean.

3.A negative z-score reveals the raw score is below the mean average. For example, if a z-score is equal to -2, it is 2 standard deviations below the mean.

4.Another way to interpret z-scores is by creating a standard normal distribution (also known as the z-score distribution or probability distribution).

The curve shown in the figure 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 .

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.

Also , the  area between µ + 1σ and  µ + 2σ is 13.6% of the total area.

 

*** Q1 solve Check the attachment for excel file .