This note is only used for learning on Hackerrank - 10 days of Statistics.

Day 0 : Mean, Median, Mod, Weighted mean

• mean = mean value $\frac{1}{n}\sum_i x_i$
• median = the number at the center, if the number of elements are odd, it’s the center number, if even, it’s the mean of two center elements.
• mod = number(s) with the most number of appearances.
• Given a set of numbers, X, and corresponding set of weights, W, the weighted mean is calculated as follows $m_w = \dfrac{\Sigma (x_i\times w_i)}{\Sigma w_i}$

• Find median without numpy

    def find_median(lst):
        len_lst = len(lst)
        if len_lst % 2 == 1:
                return lst[len_lst//2]
        else:
                return (lst[len_lst//2-1] + lst[len_lst//2])/2
  

• With numpy

    import numpy as np
    from scipy import stats 
  	
    print(np.mean(<list>))
    print(np.median(<list>))
    print(int(stats.mode(<list>)[0]))
  

Day 1 : Quartiles, Interquartile Range, Standard Deviation

• Quartile of an ordered data set are the 3 points that split the data set into 4 groups.
  • $Q_1$: the middle number between the smallest number in a data set and its median
  • $Q_2$: the median ($50^{th}$ percentile) of the data set
  • $Q_3$: the middle number between a data set’s median and its largest number
  • Algorithm:
    • If the number of elements is odd, don’t include the median for each half when seeking $Q_1, Q_2$
    • If the number of elements is even, just devide into 2 halves.
    • $Q_1$ is the median of first half, $Q_2$ is the median of second half.
• Get the input:

    size = int(input())
    numbers = list(map(int, input().split()))
  

• The interquartile range of an array is the difference between its first (Q1) and third (Q3) quartiles
• Output (0.9): print("{:.1f}".format(Q3-Q1))
• Expected value $\mu$ = mean of discrete random variable X.

• Variance $\sigma^2$: $\sigma^2 = \dfrac{\Sigma (x_i-\mu)}{n}$

• Standard deviation $\sigma$: $\sigma^2 = \sqrt{\dfrac{\Sigma (x_i-\mu)}{n}}$

• Use sqrt: import math as m

Day 2 : Basic probability

• $P(A) = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}$
• $0\le P(A)\le 1$
• $P(A^C) = 1-P(A)$
• mutually exclusive or disjoint: $A\cap B=\emptyset, P(A\cap B)=0$
• A,B are disjoint (mutually exclusive): $P(A\cup B) = P(A) + P(B)$
• A,B are independent: $P(A \cap B) = P(A)\times P(B)$
• Genreal: $\vert A\cup B\vert = \vert A\vert + \vert B\vert -\vert A\cap B\vert$

Day 3 : Conditional probability

• A, B are considered to be independent if event: $P(B\vert A) = P(B)$
• $P(A\cap B) = P(B\vert A)\times P(A)$
• $P(B\vert A) = \dfrac{P(A\cap B)}{P(A)}$
• Bayes’ theorem:

$$P(A\vert B) = \dfrac{P(B\vert A)\times P(A)}{P(B)} = \dfrac{P(B\vert A)\times P(A)}{P(B\vert A)\times P(A) + P(B\vert A^c)\times P(A^c)}$$

• Permutations (hoán vị): take r-element permutation from n elements (don’t care about order): $nPr = \dfrac{n!}{(n-r)!}$
• Combinations (chỉnh hợp): take r-element from n elements (care about order): $nCr = \dfrac{nPr}{r!} = \dfrac{n!}{r!(n-r)!}$

Day 4 : Binomial distribution

• Tutorial link.
• Random variable X:is the real value function $X: S\to R$ in which there is an event for each interval $I\subseteq R$
• A binomial experiment (or Bernoulli trial) is a statistical experiment that has the following properties:
  • The experiment consists of repeated trials.
  • The trials are independent.
  • The outcome of each trial is either success ($s$) or failure ($f$).
• Bernoulli Random Variable and Distribution: The sample space of a binomial experiment only contains two points, s and f. We define a Bernoulli random variable to be the random variable defined by $X(s)=1$ and $X(f)=0$. If we consider the probability of success to be p and the probability of failure to be q (where $q=1-p$), then the probability mass function (PMF) of is:

$$p(x) = \begin{cases} 1-p=q \quad if\, x=0,\\ p \quad if\, x=1\\ 0 \quad otherwise \end{cases}$$

or

$$f(x) = p^x(1-p)^{1-x},\quad for\, x\in \{0,1\}.$$

• Binomial distribution: is the binomial probability, meaning the probability of having exactly x successes out of n trials.
  • The number of successes is x.
  • The total number of trials is n.
  • The probability of success of 1 trial is p.
  • The probability of failure of 1 trial q, where q=1-p.

$$b(x,n,p) = \binom {n}{x}\times p^x \times q^{1-x}$$

• Cumulative Probability (CDF): $F_X(x) = P(X\le x)$ and $P(a<X\le b) = F_X(b)-F_X(a)$.

Day 4 : Geometric Distribution

• Negative Binomial Experiment: A negative binomial experiment is a statistical experiment that has the following properties:
  • The experiment consists of n repeated trials.
  • The trials are independent.
  • The outcome of each trial is either success (s) or failure (f).
  • P(s) is the same for every trial.
  • The experiment continues until x successes are observed.
• If x is the number of experiments until the xth success occurs, then X is a discrete random variable called a negative binomial.

$$b^{\ast}(x,n,p) = \binom {n-1}{x-1}\times p^x \times q^{1-x}$$

• The number of successes to be observed is x.
• The total number of trials is n.
• The probability of success of 1 trial is p.
• The probability of failure of 1 trial q, where q=1-p.

• $b^{\ast}(x,n,p)$ is the negative binomial probability, meaning the probability of having x-1 successes after n-1 trials and having x successes after n trials.

• Geometric Distribution: The geometric distribution is a special case of the negative binomial distribution that deals with the number of Bernoulli trials required to get a success (i.e., counting the number of failures before the first success). Recall that X is the number of successes in n independent Bernoulli trials, so for each i (where $1\le i \le n$):

$$X_i = \begin{cases} 1 \quad \text{if the ith trial is a success}\\ 0 \quad \text{otherwise} \end{cases}$$

• The geometric distribution is a negative binomial distribution where the number of successes is 1. We express this with the following formula:

$$g(n,p) = q^{n-1}\times p.$$

Day 5 : Poisson Distribution

• A Poisson experiment is a statistical experiment that has the following properties:
  • The outcome of each trial is either success or failure.
  • The average number of successes ($\lambda$) that occurs in a specified region is known.
  • The probability that a success will occur is proportional to the size of the region.
  • The probability that a success will occur in an extremely small region is virtually zero.
• A Poisson random variable is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a Poisson distribution:

$$P(k, \lambda) = \dfrac{\lambda^k e^{-\lambda}}{k!}$$

• $e=2.71828$
• $\lambda$ is the average number of successes that occur in a specified region.
• k is the actual number of successes that occur in a specified region.
• $P(k,\lambda)$ is the Poisson probability, which is the probability of getting exactly k successes when the average number of successes is $\lambda$.
• Check examples & special case here.
• Special case: X has Poisson Distribution, $E[X] = \lambda = Var(X), Var(X) = E[X^2] - (E[X])^2$

Day 5 : Normal Distribution

• The probability density of normal distribution is:

$$N(\mu,\sigma^2) = \dfrac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

• $\mu$ is the mean (or expectation) of the distribution. It is also equal to median and mode of the distribution.
• $\sigma^2$ is the variance.
• $\sigma$ is the standard deviation.
• In python, we can use numpy.random.normal (ref)
• If $\mu=0, \sigma=1$ then the normal distribution is known as standard normal distribution

$$\phi(x) = \dfrac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}$$

• Every normal distribution can be represented as standard normal distribution:

$$N(\mu,\sigma^2) = \frac{1}{\sigma}\phi(\frac{x-\mu}{\sigma})$$

• Consider a real-valued random variable, X. The cumulative distribution function of X (or just the distribution function of X) evaluated at x is the probability that X will take a value less than or equal to x:

$$F_X(x) = P(X\le x) \\ P(a\le X\le b) = P(a<X<b) = F_X(b) - F_X(a)$$

• The cumulative distribution function for a function with normal distribution is (erf = error function):

$$\Phi(x) = \frac{1}{2}(1+\text{erf}(\frac{x-\mu}{\sigma\sqrt{2}})) \\ \text{erf}(z) = \frac{2}{\sqrt{\pi}}\int_0^z e^{-x^2}dx$$

• In python, we can use math.erf function.