Basics of ML & AL Week 2 Challenge
Q1) To calculate the business moments (Mean, Variance, Skewness, and Kurtosis) for the given data set, we will follow these steps:
1. Mean (μ): The sum of the product of each value of X and its corresponding probability, P(x).
2. Variance (σ²): The sum of the squared difference between each value of X and the mean, multiplied by their respective probabilities, P(x).
3. Skewness (γ₁): The sum of the cubed difference between each value of X and the mean, divided by the cube of the standard deviation, and multiplied by their respective probabilities, P(x).
4. Kurtosis (γ₂): The sum of the fourth power of the difference between each value of X and the mean, divided by the square of the standard deviation's fourth power, and multiplied by their respective probabilities, P(x).
Given Data Set:
X(Random Variable) = Number of toys
P(x) = Probability of selling X toys in a day
0 0.35
1 0.25
2 0.15
3 0.15
4 0.10
Step 1: Calculate the Mean (μ)
μ = (0 * 0.35) + (1 * 0.25) + (2 * 0.15) + (3 * 0.15) + (4 * 0.10)
μ = 0 + 0.25 + 0.30 + 0.45 + 0.40
μ = 1.40
Step 2: Calculate the Variance (σ²)
σ² = [(0 - 1.40)² * 0.35] + [(1 - 1.40)² * 0.25] + [(2 - 1.40)² * 0.15] + [(3 - 1.40)² * 0.15] + [(4 - 1.40)² * 0.10]
σ² = (0.96 * 0.35) + (0.09 * 0.25) + (0.29 * 0.15) + (1.44 * 0.15) + (1.96 * 0.10)
σ² = 0.336 + 0.0225 + 0.0435 + 0.216 + 0.196
σ² = 0.797
Step 3: Calculate the Skewness (γ₁)
First, calculate the standard deviation (σ): σ = √σ² = √0.797 ≈ 0.894
γ₁ = [(0 - 1.40)³ / 0.894³] *
Q2) Expected value adds more depth to the concept of central tendency by incorporating probabilities. It represents the long-term average of an experiment or process when repeated multiple times. This helps in understanding the likely outcome under different scenarios, which is crucial in decision-making and risk assessment. While the mean gives you the simple average, expected value considers the probability distribution of the outcomes, making it more useful in various applications like finance, economics, and statistics.
Q3) In statistical analysis, skewness refers to the asymmetry of a data distribution. A curve with skewness is not perfectly symmetrical around the mean. While it is generally preferred to have a symmetrical distribution, it is not always bad. The impact of skewness depends on the context and the type of data being analyzed. If the skewness significantly affects the interpretation of your results or the applicability of certain statistical tests, then it may be considered a problem. However, in some cases, skewness can provide valuable insights and still allow for meaningful analysis.