Basics of ML & AL Week 2 Challenge

1) Calculate all 4 business moments using the given data (First Image)

The four business moments are:

1. Mean (First Moment)
2. Variance (Second Moment)
3. Skewness (Third Moment)
4. Kurtosis (Fourth Moment)

We will calculate them using the given random variable X and probabilities P(X).

Step 1: Calculate Mean (Expected Value, E[X])

The mean or expected value $E[X]$E[X] is given by:

$$E[X] = \sum X_i P(X_i)$$E[X]=∑Xi​P(Xi​)

Using the given data:

$$E[X] = (0 \times 0.35) + (1 \times 0.25) + (2 \times 0.15) + (3 \times 0.15) + (4 \times 0.10)$$E[X]=(0×0.35)+(1×0.25)+(2×0.15)+(3×0.15)+(4×0.10) $$E[X] = 0 + 0.25 + 0.30 + 0.45 + 0.40 = 1.4$$E[X]=0+0.25+0.30+0.45+0.40=1.4

Step 2: Calculate Variance (Second Moment)

Variance is given by:

$$Var(X) = E[X^2] - (E[X])^2$$Var(X)=E[X2]−(E[X])2

First, calculate $E[X^2]$E[X2]:

$$E[X^2] = (0^2 \times 0.35) + (1^2 \times 0.25) + (2^2 \times 0.15) + (3^2 \times 0.15) + (4^2 \times 0.10)$$E[X2]=(02×0.35)+(12×0.25)+(22×0.15)+(32×0.15)+(42×0.10) $$E[X^2] = 0 + 0.25 + 0.60 + 1.35 + 1.60 = 3.8$$E[X2]=0+0.25+0.60+1.35+1.60=3.8

Now, compute variance:

$$Var(X) = 3.8 - (1.4)^2 = 3.8 - 1.96 = 1.84$$Var(X)=3.8−(1.4)2=3.8−1.96=1.84

Step 3: Calculate Skewness (Third Moment)

Skewness measures the asymmetry of the distribution:

$$Skewness = \frac{E[(X - \mu)^3]}{\sigma^3}$$Skewness=σ3E[(X−μ)3]​

Using the computed values of mean ($\mu = 1.4$μ=1.4) and standard deviation ($\sigma = \sqrt{1.84} \approx 1.356$σ=1.84​≈1.356), we calculate:

$$E[(X - \mu)^3] = \sum (X_i - \mu)^3 P(X_i)$$E[(X−μ)3]=∑(Xi​−μ)3P(Xi​) $$= (-1.4)^3 \times 0.35 + (-0.4)^3 \times 0.25 + (0.6)^3 \times 0.15 + (1.6)^3 \times 0.15 + (2.6)^3 \times 0.10$$=(−1.4)3×0.35+(−0.4)3×0.25+(0.6)3×0.15+(1.6)3×0.15+(2.6)3×0.10 $$= (-2.744) \times 0.35 + (-0.064) \times 0.25 + (0.216) \times 0.15 + (4.096) \times 0.15 + (17.576) \times 0.10$$=(−2.744)×0.35+(−0.064)×0.25+(0.216)×0.15+(4.096)×0.15+(17.576)×0.10 $$= -0.96 - 0.016 + 0.032 + 0.614 + 1.758 = 1.428$$=−0.96−0.016+0.032+0.614+1.758=1.428

Now, Skewness:

$$Skewness = \frac{1.428}{(1.356)^3} = \frac{1.428}{2.49} = 0.573$$Skewness=(1.356)31.428​=2.491.428​=0.573

Since skewness > 0, the distribution is positively skewed (right-skewed).

Step 4: Calculate Kurtosis (Fourth Moment)

Kurtosis measures the "tailedness" of the distribution:

$$Kurtosis = \frac{E[(X - \mu)^4]}{\sigma^4}$$Kurtosis=σ4E[(X−μ)4]​ $$E[(X - \mu)^4] = (-1.4)^4 \times 0.35 + (-0.4)^4 \times 0.25 + (0.6)^4 \times 0.15 + (1.6)^4 \times 0.15 + (2.6)^4 \times 0.10$$E[(X−μ)4]=(−1.4)4×0.35+(−0.4)4×0.25+(0.6)4×0.15+(1.6)4×0.15+(2.6)4×0.10 $$= (3.8416 \times 0.35) + (0.0256 \times 0.25) + (0.1296 \times 0.15) + (6.5536 \times 0.15) + (45.6976 \times 0.10)$$=(3.8416×0.35)+(0.0256×0.25)+(0.1296×0.15)+(6.5536×0.15)+(45.6976×0.10) $$= 1.34456 + 0.0064 + 0.01944 + 0.98304 + 4.56976 = 6.923$$=1.34456+0.0064+0.01944+0.98304+4.56976=6.923

Now, compute Kurtosis:

$$Kurtosis = \frac{6.923}{(1.356)^4} = \frac{6.923}{3.68} = 1.88$$Kurtosis=(1.356)46.923​=3.686.923​=1.88

Since kurtosis < 3, this is a platykurtic distribution (flatter than normal distribution).

---

2) What is the significance of expected value when simple mean is already in place?

The expected value (E[X]) is a probability-weighted mean rather than just an arithmetic mean. It is more representative when:

• Data points are assigned different probabilities (unlike the arithmetic mean, which assumes equal weight).
• It helps in decision-making under uncertainty, particularly in risk analysis, insurance, and finance.

---

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

Not necessarily. The impact of skewness depends on the type of data and context:

• In financial data, skewness can indicate high-risk investments.
• In normal distributions, extreme skewness affects statistical tests like t-tests and regression.
• In machine learning, skewness in features may lead to biased models, requiring transformation (log/square root).

---

4) Evaluate probabilities for normal distribution with $\mu = 680$μ=680, $\sigma = 31$σ=31

We use the Z-score formula:

$$Z = \frac{X - \mu}{\sigma}$$Z=σX−μ​

a) $P(X < 711)$P(X<711)

$$Z = \frac{711 - 680}{31} = \frac{31}{31} = 1$$Z=31711−680​=3131​=1

From the normal table, $P(Z < 1) = 0.8413$P(Z<1)=0.8413.

b) $P(X > 740)$P(X>740)

$$Z = \frac{740 - 680}{31} = \frac{60}{31} = 1.94$$Z=31740−680​=3160​=1.94

From the normal table, $P(Z < 1.94) = 0.9738$P(Z<1.94)=0.9738.

$$P(X > 740) = 1 - 0.9738 = 0.0262$$P(X>740)=1−0.9738=0.0262

c) $P(600 < X < 700)$P(600<X<700)

For $P(X < 700)$P(X<700):

$$Z = \frac{700 - 680}{31} = \frac{20}{31} = 0.645$$Z=31700−680​=3120​=0.645

From the normal table, $P(Z < 0.645) = 0.7400$P(Z<0.645)=0.7400.

For $P(X < 600)$P(X<600):

$$Z = \frac{600 - 680}{31} = \frac{-80}{31} = -2.58$$Z=31600−680​=31−80​=−2.58

From the normal table, $P(Z < -2.58) = 0.0049$P(Z<−2.58)=0.0049.

$$P(600 < X < 700) = 0.7400 - 0.0049 = 0.7351$$P(600<X<700)=0.7400−0.0049=0.7351

d) $P(X = 720)$P(X=720)

Since normal distribution is continuous, $P(X = 720) = 0$P(X=720)=0.

---

5) Explanation of the Curve (Second Image)

The second image represents a standard normal distribution (bell-shaped curve):

• Mean = 0, Standard Deviation = 1
• The area under the curve represents probabilities.
• The Empirical Rule states:
  • 68.27% of the data falls within 1 standard deviation ($\pm 1\sigma$±1σ).
  • 95.45% falls within 2 standard deviations ($\pm 2\sigma$±2σ).
  • 99.73% falls within 3 standard deviations ($\pm 3\sigma$±3σ).

This curve is used for statistical inference, hypothesis testing, and probability calculations.