Understanding Binomial Distribution Skewness

The question asks us to find the skewness of a random variable X that follows a binomial distribution with specific parameters. A binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

The parameters of the binomial distribution are:

• \(n\): the number of trials (given as 6)
• \(p\): the probability of success on a single trial (given as \(\dfrac{1}{4}\))

We need to calculate the skewness of this distribution. Skewness is a measure of the asymmetry of the probability distribution. For a binomial distribution, the formula for the coefficient of skewness (\(\gamma_1\)) is:

$$\gamma_1 = \frac{1 - 2p}{\sqrt{np(1-p)}}$$

Let's break down the calculation using the given parameters:

• \(n = 6\)
• \(p = \dfrac{1}{4}\)

First, calculate \(1-p\):

$$1 - p = 1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}$$

Now, calculate the term inside the square root in the denominator, which is the variance (\(\sigma^2\)) of the binomial distribution:

$$np(1-p) = 6 \times \frac{1}{4} \times \frac{3}{4}$$ $$np(1-p) = \frac{6 \times 1 \times 3}{4 \times 4} = \frac{18}{16}$$ $$np(1-p) = \frac{9}{8}$$

Next, calculate the standard deviation (\(\sigma\)), which is the square root of the variance:

$$\sqrt{np(1-p)} = \sqrt{\frac{9}{8}}$$ $$\sqrt{np(1-p)} = \frac{\sqrt{9}}{\sqrt{8}} = \frac{3}{\sqrt{4 \times 2}} = \frac{3}{2\sqrt{2}}$$

Now, let's calculate the numerator of the skewness formula:

$$1 - 2p = 1 - 2 \times \frac{1}{4}$$ $$1 - 2p = 1 - \frac{2}{4} = 1 - \frac{1}{2} = \frac{1}{2}$$

Finally, substitute the numerator and denominator values into the skewness formula:

$$\gamma_1 = \frac{1 - 2p}{\sqrt{np(1-p)}} = \frac{\dfrac{1}{2}}{\dfrac{3}{2\sqrt{2}}}$$

To divide by a fraction, we multiply by its reciprocal:

$$\gamma_1 = \frac{1}{2} \times \frac{2\sqrt{2}}{3}$$ $$\gamma_1 = \frac{1 \times 2\sqrt{2}}{2 \times 3} = \frac{2\sqrt{2}}{6}$$ $$\gamma_1 = \frac{\sqrt{2}}{3}$$

The skewness of the binomial distribution with \(n=6\) and \(p=\dfrac{1}{4}\) is \(\dfrac{\sqrt{2}}{3}\).

Revision Table: Binomial Distribution Properties

Additional Information: Understanding Skewness

• What is Skewness? Skewness is a measure of the asymmetry in a data set's distribution. A symmetric distribution, like the normal distribution, has a skewness of zero.
• Types of Skewness:
  • Positive Skewness: The tail on the right side of the distribution is longer or fatter than the tail on the left side. This indicates that there are more extreme values on the higher end. For a binomial distribution, this occurs when \(p < 0.5\).
  • Negative Skewness: The tail on the left side of the distribution is longer or fatter than the tail on the right side. This indicates more extreme values on the lower end. For a binomial distribution, this occurs when \(p > 0.5\).
  • Zero Skewness: The distribution is symmetric. For a binomial distribution, this occurs when \(p = 0.5\).
• In this problem, \(p = \dfrac{1}{4} = 0.25\), which is less than 0.5. Therefore, we expect the distribution to be positively skewed, meaning the calculated skewness value should be positive. Our result \(\dfrac{\sqrt{2}}{3}\) is indeed positive, which aligns with the expected shape for \(p < 0.5\).