Basics of ML & AL Week 2 Challenge
Q.1- Calculate all 4 business moments using pen and paper for the below data set?
| variance | skewness | kurtosis | |||
| X | P(X) | X*P(X) | ((x-mean)^2*p(x)) | ((x-mean)^3*p(x))/variance^(3/2) | (((x-mean)^4*p(x))/variance^2)-3 |
| 0 | 0.35 | 0 | 0.686 | -0.96 | 1.34 |
| 1 | 0.25 | 0.25 | 0.040 | -0.02 | 0.01 |
| 2 | 0.15 | 0.3 | 0.054 | 0.03 | 0.02 |
| 3 | 0.15 | 0.45 | 0.384 | 0.61 | 0.98 |
| 4 | 0.1 | 0.4 | 0.676 | 1.76 | 4.57 |
| sum | 1.4 | 1.84 | 1.43 | 6.923 | |
| Mean | variance | 2.50 | 3.39 | ||
| 0.572 | 2.04 | ||||
| skewness | -0.96 | ||||
| kurtosis | |||||
Mean-1.4
variance- 1.84
Skewness - 0.572
Kurtosis- -0.96
Q.2- What is the significance of expected value when simple mean (Sum of all observations/number of observations) is already in place
**Expected Value vs. Simple Mean:**
1. **Expected Value:**
- **Calculation:** It is the weighted average of all possible outcomes, considering their probabilities.
- **Formula:** \( E(X) = x_1 \cdot P(X=x_1) + x_2 \cdot P(X=x_2) + \ldots + x_n \cdot P(X=x_n) \)
- **Significance:** Provides a measure of the center of a probability distribution, crucial in decision theory and risk assessment.
2. **Simple Mean (Arithmetic Mean):**
- **Calculation:** The sum of all observations divided by the number of observations.
- **Formula:** \( \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} \)
- **Application:** A straightforward measure of average, treating all observations equally.
**Comparison:**
- **Equal Probabilities:** In some cases, especially with equally likely outcomes, the expected value may be the same as the simple mean.
- **Weighted Average:** Expected value considers the probabilities, providing a weighted average based on the likelihood of each outcome.
- **Significance:** Expected value is significant in scenarios with different probabilities, offering a nuanced measure of central tendency.
**Conclusion:**
- While the simple mean is a basic average, the expected value incorporates probabilities, making it more suitable for situations where outcomes have different likelihoods.
- The expected value's significance lies in its ability to account for uncertainty, influencing decision-making in various fields, including finance and risk assessment.
Q.3- Having skewness in the curve considered to be bad in the analysis?
*Skewness in Data Analysis:**
1. **Nature of Skewness:**
- **Positive Skewness:** Right tail is longer, suggesting potential for extreme positive values.
- **Negative Skewness:** Left tail is longer, suggesting potential for extreme negative values.
2. **Interpretation:**
- **Not Inherently Bad:** Skewness is not inherently negative; it provides insights into the distribution's asymmetry.
- **Context Matters:** Interpret skewness based on the specific analysis and characteristics of the data.
3. **Considerations:**
- **Normal Distribution:** In a perfect normal distribution, skewness is zero; real-world data often deviates.
- **Outliers Influence:** Skewness can be influenced by outliers; identifying and understanding outliers is crucial.
4. **Significance in Analysis:**
- **Financial Analysis:** Skewness can be crucial in finance where extreme values may have significant consequences.
- **Risk Management:** Skewed distributions may impact risk assessment and decision-making.
5. **Conclusion:**
- **Neutral Measure:** Skewness itself is not inherently bad; its significance depends on the analysis context.
- **Tool for Understanding:** It's a tool for understanding the distribution's shape and identifying potential outliers.
- **Adjustments Possible:** In some cases, adjustments or transformations may be considered if skewness interferes with analysis objectives.
Q.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)
Certainly! Let's calculate the probabilities using Z-values:
**a) \( P(X < 711) \):**
\[ Z = \frac{{711 - 680}}{{31}} \approx 1 \]
Using a Z-table or calculator, the probability is approximately 0.8413.
**b) \( P(X > 740) \):**
\[ Z = \frac{{740 - 680}}{{31}} \approx 1.94 \]
Using a Z-table or calculator, the probability is approximately 0.9732.
**c) \( P(600 < X < 720) \):**
\[ Z_{600} = \frac{{600 - 680}}{{31}} \approx -2.58 \]
\[ Z_{720} = \frac{{720 - 680}}{{31}} \approx 1.29 \]
Using a Z-table or calculator:
- \( P(Z < -2.58) \approx 0.0049 \)
- \( P(Z < 1.29) \approx 0.9015 \)
Therefore, \( P(600 < X < 720) \approx 0.9015 - 0.0049 \approx 0.8966 \).
**d) \( P(X = 720) \):**
For a continuous distribution, the probability of a single point is technically zero.
Q .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.