Binomial Parameters

The binomial mean

• For a binomial variable, the mean has a beautifully simple formula:
• $$\mu_X = np$$
• Trials times success-probability — the expected number of successes.
• Flip a coin $100$ times: expect $\mu_X = 100 \times 0.5 = 50$ heads.

The binomial standard deviation

• The standard deviation is:
• $$\sigma_X = \sqrt{np(1-p)}$$
• It measures how much the count of successes typically varies.
• These two parameters 参数 ($\mu_X$, $\sigma_X$) summarize the whole binomial distribution.

The binomial distribution, with mean n times p

The shape

• The shape depends on $p$: at $p = 0.5$ the distribution is symmetric.
• $p < 0.5$ → skewed right; $p > 0.5$ → skewed left.
• As $n$ grows, the shape becomes more symmetric and bell-like (even for skewed $p$).
• This near-normal shape for large $n$ powers the inference in later units.

Interpret in context

• Always state the parameters in the problem's own words.
• $\mu_X$: "we expect about $np$ successes on average."
• $\sigma_X$: "the number of successes typically varies by about $\sigma_X$."
• Numbers mean little until tied back to the real situation.

Write from a binomial mean of 3

• There are 9 independent trials.
• The success probability is 1/3. The lesson n of 20 is another setting.