06 Binomial Distribution

The binomial distribution models the number of successes in multiple independent Bernoulli trials. So if :

• You repeat a Bernoulli experiment n times
• Each trial has a success probability $\theta$
• And you want to know: how many successes (like black balls, heads, correct answer etc.) you get out of those n trials

Notation:

We write:

$$X\sim \text{Binomial}(n,\theta )$$

• $X$ : number of successes in $n$ trials
• $n$ : number of trails
• $\theta$ : probability of success in each trial
• $X\in \{0,1,2,\dots ,n\}$

Probability Mass Function (PMF) :

$$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right){\theta }^k(1-\theta {)}^{n-k}$$

Where:

• $k$ : number of successes.
• $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ : “n choose k” = number of ways to choose $k$ successes out of $n$ trials.

Key Properties:

• Mean : $\mathbb{E}[X]=n\theta$
• Variance : $\text{Var}(X)=n\theta (1-\theta )$

Real-life example:

You flip a coin 10 times (so $n$ = 10), and the probability of heads is $\theta$ = 0.6. What’s the probability of getting exactly 7 heads ?

$$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right){\theta }^k(1-\theta {)}^{n-k}$$

Where:

• $(n=10)$ (the number of trials),
• $(k=7)$ (the number of successes — in this case, heads),
• $(\theta =0.6)$ (probability of heads),
• $(1-\theta =0.4)$ (probability of tails).

$$P(X=7)=\left(\genfrac{}{}{0ex}{}{10}{7}\right)(0.6{)}^7(0.4{)}^3=0.215$$