Random variables and distributions: Data Science, Machine Learning and AI (Beginner: data foundations)

Why distributions?

A restaurant does not know exactly how many orders will come at 8 p.m., but it knows the pattern: usually 40 to 60, rarely more than 80. A distribution is that pattern written down. Once you know the distribution, you can plan staff, stock and servers, and you can tell when something unusual is happening.

Random variables

A random variable is a number whose value depends on chance. It can be discrete (it takes separate values you can count, like 0, 1, 2 customers) or continuous (any value in a range, like a delivery time of 23.7 minutes). For discrete variables, each value has a probability, and all the probabilities add up to 1.

Expected value: the long-run average

The expected value E[X] is the average you would get if you repeated the process many times. Multiply each value by its probability and add them up. For a die: (1 + 2 + 3 + 4 + 5 + 6) × 1/6 = 3.5. You never roll 3.5, but over many rolls the average gets close to it.

python

values = [-10, 990]       # lose ₹10 or win ₹990
probs = [0.999, 0.001]
ev = sum(v * p for v, p in zip(values, probs))
print(round(ev, 2))

A ₹10 ticket with a 1 in 1,000 chance to win ₹1,000 (₹990 profit). Expected value: -9.0. On average you lose ₹9 per ticket.

Four shapes you will meet

Distribution	What it counts or measures	Example
Uniform	Every value in a range is equally likely	A fair die; a random number between 0 and 1
Binomial (n, p)	Number of successes in n yes/no tries, each with chance p	How many of 10 calls lead to a sale when p = 0.2
Poisson (λ)	Number of events in a fixed time when they happen independently at an average rate λ	Calls to a helpline per minute; accidents on a road per day
Normal (μ, σ)	A continuous bell curve around the mean μ with spread σ	Heights, measurement errors, averages of many values

The binomial formula

For k successes out of n tries: P(k) = C(n, k) × p^k × (1 − p)^(n − k). C(n, k), read "n choose k", counts the ways to pick which k tries succeed. Python has it as math.comb(n, k). The expected number of successes is n × p.

python

import math

n, p = 10, 0.5
for k in [0, 5, 10]:
    print(k, round(math.comb(n, k) * p**k * (1 - p)**(n - k), 4))

Shows 0 0.001, then 5 0.2461, then 10 0.001. Five heads in ten tosses is the most likely result, but it happens only about a quarter of the time.

The Poisson formula

If events arrive at an average rate λ (lambda) per period, P(k events) = λ^k × e^(−λ) ÷ k!. A helpline that gets 3 calls a minute on average has P(0 calls) = e^(−3) ≈ 0.05 and P(2 calls) ≈ 0.224.

Tools you will use later

python

from scipy import stats

print(stats.binom.pmf(3, 10, 0.5))   # 0.1171875
print(stats.poisson.pmf(2, 3))        # about 0.224
print(stats.norm.cdf(1.96))           # about 0.975

scipy.stats has every common distribution. pmf gives P(X = k); cdf gives P(X ≤ x). Shown only; the tasks use the standard library.

Common mistakes

Using the binomial when the tries are not independent or p changes. Fix: check that each try is like the last.
Using the normal for counts that are small or cannot be negative, like 0 to 3 complaints a day. Fix: use Poisson or binomial.
Thinking the expected value is the most likely value. A lottery's expected value is a loss even though the most likely result is "lose ₹10".

Random variables and distributions

Explanation

Why distributions?