Statistics

Probability Basics

• Probability of an Event:

$$P(\text{event}) = \frac{\text{\# of ways event can happen}}{\text{total \# of possible outcomes}}$$

• Example: Probability of heads in a coin flip = $\frac{1}{2} = 0.5$

• Independent Events: Probability of the second event isn't affected by the first.

  • Sampling with replacement = events are independent.

• Dependent Events: Probability of the second event is affected by the first.

  • Sampling without replacement = events are dependent.

Sampling in Python

• Random Sampling: DataFrame.sample()

  sales_counts.sample()

• Setting a Seed: Ensures reproducibility.

  np.random.seed(10)

Discrete Distributions

• Probability Distribution: Describes probability of each outcome.
  • Expected value of a fair die roll:

$$\text{Expected Value} = (1 \times \frac{1}{6}) + (2 \times \frac{1}{6}) + \dots + (6 \times \frac{1}{6}) = 3.5$$

• Discrete Uniform Distribution:
  • Fair Die: Equal probability for each outcome.
  • Uneven Die: Different probabilities for each outcome.

Continuous Distributions

• Continuous Uniform Distribution: Equal probability across a continuous range.

  • Example: Waiting time for a bus between 0 and 12 minutes.

• Area Under Curve = Probability:

  $$P(4 \leq \text{wait time} \leq 7) = 3 \times \frac{1}{12} = 0.25$$

Using Scipy for Distributions

• Uniform Distribution:

  from scipy.stats import uniform
  uniform.cdf(value, min, max)

  • Probability of a value less than or equal to 7: uniform.cdf(7, 0, 12)

• Binomial Distribution:

  • Binary Outcomes: Success (1) or Failure (0)
  • Single Flip:

    from scipy.stats import binom
    binom.rvs(1, 0.5, size=1)  # Result of one flip

  • Many Flips, One Time:

    binom.rvs(num_trials, prob_success, size=1)

  • Probability of k Successes:

    binom.pmf(k, n, p)

Key Terms

• Expected Value (Binomial): $E = n \times p$
• Law of Large Numbers: As sample size grows, the sample mean approaches the expected value.

Example Code

• Generating Random Numbers (Uniform Distribution):

  uniform.rvs(min, range, size=sample_size)

• Rolling a Die with Probability:

  die.sample(10, replace=True)