Understanding Skewness in Binomial Distribution

The question asks for the skewness of a random variable X which follows a binomial distribution. The binomial distribution is a discrete probability distribution that represents the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

A random variable X is binomially distributed with parameters n and p, denoted as B(n, p), where:

• n is the number of trials.
• p is the probability of success on a single trial.

In this specific problem, we are given that X is binomially distributed with parameters n = 25 and p = 0.2.

Calculating Binomial Distribution Skewness

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. For a binomial distribution B(n, p), the formula for skewness ($\gamma_1$) is given by:

\begin{equation*} \gamma_1 = \frac{1-2p}{\sqrt{np(1-p)}} \end{equation*}

Let's break down the formula components for our given parameters n = 25 and p = 0.2:

• n = 25
• p = 0.2
• $1-p = 1 - 0.2 = 0.8$
• $np(1-p)$: This is the variance of the binomial distribution. $25 \times 0.2 \times 0.8 = 5 \times 0.8 = 4$.
• $\sqrt{np(1-p)}$: This is the standard deviation of the binomial distribution. $\sqrt{4} = 2$.
• $1-2p = 1 - 2(0.2) = 1 - 0.4 = 0.6$.

Now, substitute these values into the skewness formula:

\begin{equation*} \gamma_1 = \frac{0.6}{2} = 0.3 \end{equation*}

So, the skewness of the variable X is 0.3.

Comparing with Options

Let's compare our calculated skewness value (0.3) with the given options:

Our calculated value of 0.3 matches Option 4.

Conclusion on Binomial Distribution Skewness

The skewness of a binomial distribution depends on the parameter p. If p < 0.5, the distribution is positively skewed (skewed to the right). If p > 0.5, it is negatively skewed (skewed to the left). If p = 0.5, the distribution is symmetric (skewness is 0). In this case, p = 0.2, which is less than 0.5, so we expect positive skewness, which our result 0.3 confirms.

Revision Table: Binomial Distribution Skewness

Review the key properties and formulas for binomial distribution skewness.

Additional Information on Skewness and Distributions

Skewness is the third standardized moment of a probability distribution. It indicates the direction and magnitude of a distribution's asymmetry. A positively skewed distribution has a long tail on the right side, meaning the mean is typically greater than the median. A negatively skewed distribution has a long tail on the left side, with the mean typically less than the median. Symmetric distributions, like the normal distribution or a binomial distribution with p=0.5, have a skewness of zero.

Understanding skewness helps in describing the shape of a distribution alongside measures of central tendency (mean, median, mode) and dispersion (variance, standard deviation).