Binomial distribution

The first discrete model describes the number of successes in a sequence of repeated, independent trials, each with the same probability of success (Bernoulli trials).

Property — $X\sim B(n,p)$

$X$ = “number of successes in $n$ independent trials, each with probability of success $p$”. $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}, \qquad k=0,1,\ldots,n.$ Mean $\mu = np$, variance $\sigma^2 = np(1-p)$.

Example — Ten coin tosses

I toss a fair coin $10$ times. $X$ = number of heads. $X\sim B(10, 0{,}5)$.

$P(X=3) = \binom{10}{3}(0{,}5)^3(0{,}5)^7 = 120\cdot\frac{1}{1024} \approx 0{,}117.$

$\mu = 10\cdot 0{,}5 = 5$. $\quad \sigma = \sqrt{10\cdot 0{,}5\cdot 0{,}5} = \sqrt{2{,}5}\approx 1{,}58.$

The graph of the distribution $B(10;0{,}5)$ is symmetric about the mean $\mu=5$, with the probabilities decreasing rapidly towards the extreme values.

Binomial distribution $B(10;0{,}5)$: the bars are symmetric with respect to the mean $\mu=5$.

Links

Topics: Probability distributions
Concepts: Binomial distribution · Expected value · Variance
Methods: Binomial · Expected value
Skills: Probability calculation · Using formulae
People: Jacob Bernoulli