Random variables, PMF vs. PDF, and the CDF

Key idea

A discrete random variable has a probability mass function (PMF) \(p(x)=P(X=x)\), and the probabilities add up: \[ \sum_x p(x)=1. \] A continuous random variable has a probability density function (PDF) \(f(x)\ge 0\), and the total area is \(1\): \[ \int_{-\infty}^{\infty} f(x)\,dx = 1. \] For continuous \(X\), probability comes from area: \[ P(a\le X\le b)=\int_a^b f(x)\,dx, \] and any single point has probability \(0\): \(P(X=c)=0\).

The cumulative distribution function (CDF) works for both cases: \[ F(x)=P(X\le x). \]

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Summary

• Discrete: use a PMF and sum probabilities.
• Continuous: use a PDF and integrate to get areas (probabilities).

Bernoulli trials and the binomial distribution

Key idea

A random variable \(X\) follows a binomial distribution, written \(X\sim \mathrm{Bin}(n,p)\), when:

• There are a fixed number of trials \(n\).
• Each trial is independent.
• Each trial has two outcomes (success/failure).
• The probability of success is constant, \(p\).

The probability of exactly \(k\) successes is: \[ P(X=k)=\binom{n}{k}p^k(1-p)^{n-k},\quad k=0,1,\dots,n. \]

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Summary

• Binomial model: \(X\sim\mathrm{Bin}(n,p)\) for counting successes in independent Bernoulli trials.
• Binomial formula: \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\).
• Complement rule is fast for “at least one”: \(P(X\ge 1)=1-P(X=0)\).

Mean, variance, and quick binomial probability shortcuts

Key idea

For \(X\sim\mathrm{Bin}(n,p)\), the mean and variance are: \[ \mathbb{E}[X]=np,\qquad \mathrm{Var}(X)=np(1-p). \] Two common shortcuts:

• Zero successes: \(P(X=0)=(1-p)^n\).
• At least zero successes: \(P(X\ge 0)=1\) because the smallest possible count is \(0\).

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Summary

• Binomial mean: \(\mathbb{E}[X]=np\).
• Binomial variance: \(\mathrm{Var}(X)=np(1-p)\).
• Quick facts: \(P(X=0)=(1-p)^n\) and \(P(X\ge 0)=1\).

Continuous uniform distribution on \([a,b]\)

Key idea

If \(X\sim \mathrm{Uniform}[a,b]\), then every value in \([a,b]\) is equally likely in the sense that the density is constant: \[ f(x)=\frac{1}{b-a}\quad \text{for } a\le x\le b. \] Probabilities are proportional to interval length: \[ P(c\le X\le d)=\frac{d-c}{b-a}\quad (a\le c\le d\le b). \]

• Smallest possible value: \(a\)
• Largest possible value: \(b\)
• Mean: \(\mathbb{E}[X]=\dfrac{a+b}{2}\)
• Variance: \(\mathrm{Var}(X)=\dfrac{(b-a)^2}{12}\)

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Summary

• Uniform PDF: \(f(x)=\dfrac{1}{b-a}\) on \([a,b]\).
• Uniform probability: \(P(c\le X\le d)=\dfrac{d-c}{b-a}\).
• Uniform variance: \(\mathrm{Var}(X)=\dfrac{(b-a)^2}{12}\).

Normal distribution: symmetry, mean, and standard deviation

Key idea

A normal random variable is written \(X\sim\mathcal{N}(\mu,\sigma^2)\). The curve is symmetric about \(\mu\), and for a normal distribution:

• Mean = median = mode \(=\mu\).
• Total area under the curve is \(1\).
• By symmetry: \(P(X<\mu)=P(X>\mu)=\tfrac12\).
• Because it’s continuous: \(P(X=\mu)=0\).

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Summary

• Normal distributions are symmetric about \(\mu\), so half the probability is on each side.
• Increasing \(\sigma\) makes the curve wider and flatter (but total area stays \(1\)).

Standard normal distribution and z-scores

Key idea

If \(X\sim\mathcal{N}(\mu,\sigma^2)\), we standardize with the z-score: \[ Z=\frac{X-\mu}{\sigma}. \] The standardized variable satisfies \(Z\sim\mathcal{N}(0,1)\). This lets you use the standard normal CDF \(\Phi(z)=P(Z\le z)\), symmetry (\(\Phi(-z)=1-\Phi(z)\)), and z-tables or calculators.

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Summary

• Standardize with \(z=\dfrac{x-\mu}{\sigma}\).
• The standard normal distribution is centered at \(0\), so its median is \(0\).

Choosing the right distribution and practicing mixed questions

Where these distributions show up

• Binomial: quality control (pass/fail), A/B tests (conversion), and repeated trials with success probability \(p\).
• Uniform: random selection over an interval (a random time, position, or measurement range).
• Normal: measurement error, heights, test scores, and many natural “bell-shaped” variations.

Worked example: uniform probability on \([0,9]\)

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Final recap

• Discrete vs. continuous: discrete uses a PMF (sums); continuous uses a PDF (areas/integrals).
• Binomial: \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\), \(\mathbb{E}[X]=np\), \(\mathrm{Var}(X)=np(1-p)\).
• Uniform: \(P(c\le X\le d)=\dfrac{d-c}{b-a}\), \(\mathrm{Var}(X)=\dfrac{(b-a)^2}{12}\).
• Normal symmetry: \(P(X<\mu)=P(X>\mu)=\tfrac12\) and \(P(X=\mu)=0\).
• Z-scores: \(z=\dfrac{x-\mu}{\sigma}\) to use standard normal probabilities.