Agenda

<aside> 📌 Reminders

• Next week last tutorial → OH </aside>

Central Limit Theorem (CLT)

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The Central Limit Theorem: states that the sampling distribution of the sample mean approaches a normal distribution as the sample size $n$ increases, regardless of the population's original distribution.

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• Key Points:
  • Works best when $n \geq 30$ (rule of thumb).
  • The mean of the sampling distribution equals the population mean ($\mu$).
  • The standard deviation of the sampling distribution (standard error) is $\sigma / \sqrt{n}$.

Example: Rolling a die 50 times and computing the average roll; repeating this process many times will yield a bell-shaped curve.

Negative Binomial Distribution

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The negative binomial distribution: models the number of trials $X$ needed to achieve $r$ successes in independent Bernoulli trials with success probability $p$

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• Parameters:

  • $r$: Number of successes
  • $p$: Probability of success in each trial

• Probability Mass Function (PMF):

  $$ P(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}, \quad k = r, r+1, \ldots  $$

• Mean and Variance:

  • $\mu = \frac{r}{p}$
  • $\sigma^2 = \frac{r(1-p)}{p^2}$

Example: Counting how many coin flips are needed to get 5 heads.

Connection:

The negative binomial distribution generalizes the geometric distribution (the case where $r = 1$) and is related to the binomial distribution. For large $r$, it can be approximated by a normal distribution due to the CLT.

Practice