Question

Let, random variable \(X \sim \text{Bernoulli}(p)\). Then, \(\beta_1\) is

• A. \(\frac{(1-2p)}{p(1-p)}\)
• B. \(\frac{(1-2p)^2}{p(1-p)}\)
• C. \(\frac{p(1-p)}{(1-2p)}\)
• D. \(\frac{p^2(1-p)}{(1-2p)}\)

Hint:

Remember the key moments for a Bernoulli(p) distribution: \(\mu=p\), \(\mu_2=p(1-p)\), and \(\mu_3=p(1-p)(1-2p)\). Knowing these allows for the rapid calculation of skewness (\(\gamma_1\)) and squared skewness (\(\beta_1\)). The distribution is symmetric only when \(p=0.5\), which makes \(\mu_3=0\) and thus \(\beta_1=0\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question asks for Pearson's coefficient of skewness, \(\beta_1\), for a Bernoulli distribution. This coefficient is a measure of the asymmetry of the probability distribution. It is defined in terms of the central moments of the distribution.

Step 2: Key Formula or Approach:
The coefficient of skewness \(\beta_1\) is defined as: \[ \beta_1 = \frac{\mu_3^2}{\mu_2^3} \] where \(\mu_2\) is the second central moment (the variance) and \(\mu_3\) is the third central moment. For a Bernoulli(\(p\)) distribution, \(X=1\) with probability \(p\) and \(X=0\) with probability \(1-p\). The central moments are calculated as \(\mu_k = E[(X-\mu)^k]\), where \(\mu\) is the mean.

Step 3: Detailed Explanation:
For a Bernoulli(\(p\)) distribution:

The mean is \(\mu = E[X] = 1 . p + 0 . (1-p) = p\).
The second central moment (variance) is \(\mu_2 = E[(X-p)^2] = \sigma^2 = p(1-p)\). \end{itemize} Now we need to calculate the third central moment, \(\mu_3\): \[ \mu_3 = E[(X-\mu)^3] = E[(X-p)^3] \] We sum the possible values of \((X-p)^3\) weighted by their probabilities:

If \(X=1\), the value is \((1-p)^3\) with probability \(p\).
If \(X=0\), the value is \((0-p)^3 = -p^3\) with probability \(1-p\). \end{itemize} So, \[ \mu_3 = p(1-p)^3 + (1-p)(-p^3) \] Factor out the common term \(p(1-p)\): \[ \mu_3 = p(1-p) \left[ (1-p)^2 - p^2 \right] \] Using the difference of squares formula, \(a^2 - b^2 = (a-b)(a+b)\): \[ \mu_3 = p(1-p) \left[ ((1-p)-p)((1-p)+p) \right] \] \[ \mu_3 = p(1-p) [ (1-2p)(1) ] = p(1-p)(1-2p) \] Finally, we calculate \(\beta_1\): \[ \beta_1 = \frac{\mu_3^2}{\mu_2^3} = \frac{[p(1-p)(1-2p)]^2}{[p(1-p)]^3} \] \[ \beta_1 = \frac{p^2(1-p)^2(1-2p)^2}{p^3(1-p)^3} \] Cancel out common terms: \[ \beta_1 = \frac{(1-2p)^2}{p(1-p)} \]
Step 4: Final Answer:
The coefficient of skewness \(\beta_1\) for a Bernoulli distribution is \(\frac{(1-2p)^2}{p(1-p)}\).