# Sums and Limits of Coin Throws

We explain why certain distributions arise naturally as the limit of coin throws.

• Bernoulli, Binomial Distributions, Geometric Distributions.
• Binomial to Poisson Distribution; Geometric to Exponential; Binomial to Normal.

Bernoulli random variables are just choice tosses: heads or tails; zero or one random variables. You can get surprisingly far – perhaps almost everywhere – in probability by summing and limiting sequences of these random variables.

Def [Bernoulli Random Variable] A random variable $X$ that takes at most two values is called a Bernoulli random variable. We assume (unless stated otherwise) that these values are zero or one. So

for some $p\in [0,1]$. We write $X\sim Bern(p)$

Ex 1 [Bernoulli to Binomial] Let $X_i$, $i=1,...n$ be IID Bernoulli RVs, let

Show that

Ans 1 Probability of $k$ ones in a row and $n-k$ zeros in a row is $p^k (1-p)^{n-k}$. The number of sequence with $k$ ones and $n-k$ zeros is

Def [Binomial Distribution] A RV $Y$ has a Binomial distribution, and we write $Y\sim Bin(n,p)$ when

Ex 2 [Bernoulli to Geometric] Consider a sequence of Bernoulli RVs $X_1,X_2,...$. Let $G$ be the index of the first $1$ in this sequence. Show that

Ans 2

Def [Geometric Distribution] A RV $G$ has Geometric Distribution, and we write $G \sim Geom(p)$ if

We now consider some limits of Binomial Random Variables.

Ex 3 [Binomial to Poisson]Consider a sequence of Binomial RVS: $Y^n\sim Bin (n, \frac{\lambda}{n})$ for some $\lambda >0$. Show that

Ans 3

Above the term in square brackets goes to one and $(1-\lambda/n)^n$ goes to $e^{-\lambda}$.

Def [Poisson Distribution] For parameter $\lambda>0$, a RV $N$ Poisson distribution and we write $N\sim Po(\lambda)$ if

Ex 4 [Geometric to Exponential] Consider a sequence of Geometric RVS: $X^n\sim Geom (\frac{\lambda}{n})$ for some $\lambda >0$. Show that

Ans 4

We now work to show that the sum of binomial distributions converges to a specific distribution called the normal distribution.

Thrm [Binomial to Normal] If $Y \sim Bin (2n , \frac{1}{2})$ then

Def [Normal Distribution] For mean $\mu$ and variance $\sigma^2$ we say that $Z$ has a normal distribution and write $Z\sim {\mathcal N}(\mu,\sigma^2)$ when

This is a special case of the central limit theorem, and involves several steps.

Ex 5 [Binominal to Normal] If $Y \sim Bin (2n , \frac{1}{2})$ Show that

Ans 6

Cancelling and dividing by $n^k$ gives the required result.

Ex 7 [Continued] Show that

Ans 7 Applying the approximation $\left( 1-x \right)= e^{-x +O(x^2)}$ we have that

Applying this also to the denominator gives the result.

Ex 8 Show, using Stirling’s Approximiation, that, as $n\rightarrow \infty$,

Ans 8 By Stirlings, $latex n! \sim \sqrt{2\pi n}\cdot e^{-n} n^n$,

Ex 9 [Continued] Argue that Thrm [SL:Bin2Norm] holds i.e. that

Ans 9 In our case the probability corresponds to the sum

After applying substitution $x\sqrt{\frac{n}{2}}=k$.