Basics of ML & AL Week 2 Challenge
Answer no.1
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
Answer no.2
The simple mean, which is the sum of all observations divided by the number of observations, is a measure of the central tendency of a dataset. It gives us an idea about where the data is clustered around.
On the other hand, the expected value is a concept from probability theory and statistics. It is a measure of the central tendency of a random variable, which is a variable that can take on different values with certain probabilities. The expected value is calculated by taking the sum of the product of each possible outcome with its probability.
While the simple mean can be used to describe the distribution of a sample, it is limited to situations where all observations have the same probability of occurring. The expected value, on the other hand, can be used to describe the distribution of a random variable, which may have different probabilities of taking on different values.
The expected value is particularly useful when we want to make predictions about the future based on a set of possible outcomes and their probabilities. For example, if we know the probabilities of different outcomes in a game of chance, we can use the expected value to determine the average amount we would expect to win or lose over a large number of trials.
In summary, while the simple mean is a measure of central tendency for a sample, the expected value is a measure of central tendency for a random variable and is particularly useful in situations where we want to make predictions based on probability distributions.
The presence of skewness in a distribution is not necessarily bad or good. Skewness is simply a measure of the degree of asymmetry in a distribution.
In general, a perfectly symmetric distribution has a skewness of 0. A positive skewness indicates that the distribution has a longer tail on the right side, meaning that there are more observations with high values than would be expected if the distribution were perfectly symmetric. A negative skewness indicates that the distribution has a longer tail on the left side, meaning that there are more observations with low values than would be expected if the distribution were perfectly symmetric.
Skewness can be useful in understanding the nature of the data and its distribution. However, the presence of skewness does not necessarily mean that the data is not useful or that any analysis based on it is not valid. In fact, many real-world datasets are naturally skewed due to various factors such as outliers, measurement errors, or underlying distributions.
That being said, it is important to consider the presence of skewness when choosing appropriate statistical methods for analysis. For example, if a dataset is heavily skewed, it may be more appropriate to use non-parametric statistical tests rather than parametric ones that assume normality.
Answer no.4
To evaluate the probabilities for a continuous normal distribution with mean μ = 680 and standard deviation σ = 31, we can use the standard normal distribution and convert the values using the z-score formula:
z = (x - μ) / σ
where x is the value we are interested in, μ is the mean, and σ is the standard deviation.
a) P(X < 711) To calculate P(X < 711), we first need to calculate the z-score:
z = (711 - 680) / 31 = 1
We can look up the probability of a standard normal distribution being less than 1 in a z-table or calculator, which is approximately 0.8413.
Therefore, P(X < 711) ≈ 0.8413.
b) P(X > 740) To calculate P(X > 740), we first need to calculate the z-score:
z = (740 - 680) / 31 = 1.935
We can look up the probability of a standard normal distribution being greater than 1.935 in a z-table or calculator, which is approximately 0.0269.
Therefore, P(X > 740) ≈ 0.0269.
c) P(600 < X < 660) To calculate P(600 < X < 660), we first need to calculate the z-scores for each value:
z1 = (600 - 680) / 31 = -2.58 z2 = (660 - 680) / 31 = -0.65
We can look up the probabilities of a standard normal distribution being less than these z-scores in a z-table or calculator.
P(X < 600) ≈ 0.0047 P(X < 660) ≈ 0.2580
Therefore, P(600 < X < 660) ≈ 0.2580 - 0.0047 = 0.2533
d) P(X = 720) The probability of a continuous normal distribution having a specific value is zero. Therefore, P(X = 720) = 0.
This curve shows the normal distribution of a random variable. The percentages show the standard deviations from the central value i.e. the mean. The standard deviation is a measure of dispersion and is used to determine how spread apart the values in a distribution/sample are from the mean and from each other. The greater the standard deviation, the more the values deviate from the mean and vice versa. The standard deviation is derived from the variance by finding its square root.
1 standard deviation from the mean is approximately 34.1% to either 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 plus or minus 34.1%. So, this means that approximately 68.2% of the values in the distribution fall within 1 standard deviation of the mean.
Similarly, 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% (34.1+34.1+13.6+13.6) of the values in the distribution fall within 2 standard deviations of the mean.
This continues to the third standard deviation and so on with the area under the curve increasing.
For an ideal normal distribution curve, it will have no skewness and so will have perfect symmetry. The mean of this distribution will be found at the centre of the x-axis of this distribution and corresponds to the peak of the distribution. This can be considered as the line of symmetry. For such a distribution, the measures of central tendency i.e. the mean, median and mode are all considered equal. In addition, the total area under this normal distribution curve is 1, because the summation of probabilities of an event is 1.