Continuous Distribution Exploring Mean, Variance, And Symmetry
In the realm of probability and statistics, continuous distributions play a pivotal role in modeling various real-world phenomena. This comprehensive exploration delves into the intricacies of a specific continuous distribution, characterized by the frequency density function f(x) = y₀x(2 - x), defined over the interval 0 ≤ x ≤ 2. Our journey will encompass the determination of key statistical measures, including the mean, variance, β₁ and β₂ coefficients, and the mean deviation about the mean. Furthermore, we will rigorously demonstrate the symmetrical nature of this distribution.
Deciphering the Frequency Density Function
The foundation of our analysis lies in the frequency density function, f(x) = y₀x(2 - x). This function, a cornerstone of continuous distributions, dictates the probability of a random variable falling within a specific interval. To fully grasp its essence, we must first unravel the significance of y₀. This constant, y₀, plays a crucial role in ensuring that the total probability over the entire distribution equals 1, a fundamental requirement for any valid probability distribution. Mathematically, this translates to the integral of f(x) over the interval [0, 2] being equal to 1.
Determining the Value of y₀
To embark on this crucial step, we must solve the following integral equation:
∫₀² y₀x(2 - x) dx = 1
Expanding the integrand, we get:
∫₀² y₀(2x - x²) dx = 1
Now, we can integrate term by term:
y₀ [x² - (x³/3)]₀² = 1
Evaluating the definite integral at the limits, we obtain:
y₀ [(2² - (2³/3)) - (0² - (0³/3))] = 1
Simplifying the expression, we have:
y₀ [4 - (8/3)] = 1
y₀ (4/3) = 1
Finally, solving for y₀, we find:
y₀ = 3/4
This crucial value of y₀ allows us to express the complete frequency density function as:
f(x) = (3/4)x(2 - x), 0 ≤ x ≤ 2
With the frequency density function fully defined, we are now equipped to delve into the calculation of essential statistical measures, starting with the mean.
Unveiling the Mean: The Center of the Distribution
The mean, a cornerstone of descriptive statistics, represents the average value of a random variable within a distribution. For continuous distributions, the mean, often denoted by µ, is calculated by integrating the product of the variable x and the frequency density function f(x) over the distribution's entire range. In our case, this translates to:
µ = ∫₀² x * f(x) dx
Substituting our derived frequency density function, we get:
µ = ∫₀² x * (3/4)x(2 - x) dx
Simplifying the integrand, we have:
µ = (3/4) ∫₀² x²(2 - x) dx
µ = (3/4) ∫₀² (2x² - x³) dx
Now, we can integrate term by term:
µ = (3/4) [(2x³/3) - (x⁴/4)]₀²
Evaluating the definite integral at the limits, we obtain:
µ = (3/4) [((2(2)³/3) - ((2)⁴/4)) - ((2(0)³/3) - ((0)⁴/4))]
Simplifying the expression, we have:
µ = (3/4) [(16/3) - 4]
µ = (3/4) (4/3)
Finally, we arrive at the mean:
µ = 1
Thus, the mean of this continuous distribution is 1, signifying the central tendency of the random variable within the specified interval.
Unveiling the Variance: Quantifying Data Dispersion
The variance, denoted by σ², stands as a fundamental measure of dispersion in statistics, quantifying the spread or variability of data points around the mean. For a continuous distribution, the variance is computed by integrating the squared difference between the variable x and the mean µ, weighted by the frequency density function f(x), over the distribution's range. Mathematically, this can be expressed as:
σ² = ∫₀² (x - µ)² * f(x) dx
However, a computationally more convenient formula for variance exists, often referred to as the computational formula:
σ² = ∫₀² x² * f(x) dx - µ²
This formula allows us to calculate the variance by first finding the integral of x² multiplied by the frequency density function and then subtracting the square of the mean. We already know that µ = 1, so we proceed to evaluate the integral:
∫₀² x² * f(x) dx = ∫₀² x² * (3/4)x(2 - x) dx
Simplifying the integrand:
(3/4) ∫₀² x³(2 - x) dx = (3/4) ∫₀² (2x³ - x⁴) dx
Integrating term by term:
(3/4) [(2x⁴/4) - (x⁵/5)]₀²
Evaluating the definite integral at the limits:
(3/4) [((2(2)⁴/4) - ((2)⁵/5)) - ((2(0)⁴/4) - ((0)⁵/5))]
Simplifying:
(3/4) [(8 - (32/5))]
(3/4) [(40 - 32)/5]
(3/4) (8/5) = 6/5
Now, we can compute the variance using the computational formula:
σ² = (6/5) - (1)²
σ² = 6/5 - 1
σ² = 1/5
Therefore, the variance of this continuous distribution is 1/5, providing a quantitative measure of the data's spread around the mean.
Delving into Skewness and Kurtosis: Unveiling β₁ and β₂
To gain a deeper understanding of the distribution's shape, we venture into the realm of skewness and kurtosis, quantified by the coefficients β₁ and β₂, respectively.
Skewness (β₁): Measuring Asymmetry
Skewness, captured by the coefficient β₁, reveals the asymmetry of a distribution. A symmetrical distribution exhibits a skewness of 0, while a positive skewness indicates a longer tail on the right side, and a negative skewness signifies a longer tail on the left side. β₁ is defined as:
β₁ = μ₃²/σ⁶
Where μ₃ is the third central moment, and σ is the standard deviation (the square root of the variance). The third central moment is calculated as:
μ₃ = ∫₀² (x - μ)³ * f(x) dx
Substituting our values for μ and f(x):
μ₃ = ∫₀² (x - 1)³ * (3/4)x(2 - x) dx
Expanding and simplifying the integrand can be cumbersome. However, recognizing that symmetry about the mean implies that the integral of any odd power of (x - μ) will be zero, we can anticipate that μ₃ will be 0 for this distribution. This observation stems from the fact that the positive and negative deviations from the mean will cancel each other out in a symmetrical distribution.
To confirm this, we can expand (x - 1)³ and multiply it by (3/4)x(2 - x), then integrate over the interval [0, 2]. After performing the integration (which involves polynomial integration), we indeed find that:
μ₃ = 0
Therefore, β₁ becomes:
β₁ = 0² / (1/5)³ = 0
A β₁ of 0 strongly suggests that the distribution is symmetrical.
Kurtosis (β₂): Gauging the Tail's Heaviness
Kurtosis, represented by β₂, measures the
