Discrete & Continuous Distributions I Practice Quiz with a Step-by-Step Interactive Lesson
Use the question set below to practice the core ideas of discrete and continuous probability distributions. This theme focuses on the most common foundations you need for statistics and probability: random variables and distribution language, discrete vs. continuous distributions, probability mass functions (PMF), probability density functions (PDF), and the cumulative distribution function (CDF), the binomial distribution \(\mathrm{Bin}(n,p)\) with the binomial formula \(\binom{n}{k}p^k(1-p)^{n-k}\), quick probability techniques like the complement rule, mean and variance formulas such as \(\mathbb{E}[X]=np\) and \(\mathrm{Var}(X)=np(1-p)\), the continuous uniform distribution \(\mathrm{Uniform}[a,b]\) with interval probabilities, and the normal distribution \(\mathcal{N}(\mu,\sigma^2)\) including symmetry, area-under-the-curve meaning, and z-scores \(z=\dfrac{x-\mu}{\sigma}\). If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks.
Answer the question set and review your mistakes at the end.
How this distributions practice works
1. Take the practice set: answer the discrete and continuous distributions questions below.
2. Open the lesson (optional): review PMF/PDF/CDF, binomial probabilities, uniform interval probabilities, and normal distribution symmetry with clear examples.
3. Retry: return to the question set and apply the distribution rules immediately.
What you will learn in the Discrete & Continuous Distributions I lesson
Random variables & distribution functions
Discrete vs. continuous random variables (counting outcomes vs. measuring on an interval)
PMF vs. PDF, why \(\sum p(x)=1\) and \(\int f(x)\,dx=1\), and why \(P(X=c)=0\) for continuous \(X\)
CDF \(F(x)=P(X\le x)\) and how it packages probabilities
Discrete distributions: Bernoulli & binomial
Binomial conditions: fixed \(n\), independent trials, two outcomes, constant \(p\)
Purpose: Build a clear understanding of discrete and continuous probability distributions so you can classify random variables, use PMF, PDF, and CDF language correctly, compute probabilities for binomial distributions and continuous uniform distributions, and interpret the normal distribution using symmetry and z-scores.
Success criteria
Distinguish discrete vs. continuous random variables.
Use a PMF \(p(x)=P(X=x)\) for discrete distributions and know \(\sum p(x)=1\).
Use a PDF \(f(x)\) for continuous distributions and know \(\int_{-\infty}^{\infty} f(x)\,dx=1\).
Explain why for continuous \(X\), \(P(X=c)=0\) for any single value \(c\).
Use the CDF \(F(x)=P(X\le x)\) to package probabilities.
Identify binomial settings and apply \(X\sim\mathrm{Bin}(n,p)\).
Use the binomial formula \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\).
Continuous: takes values on an interval (like times, lengths, weights).
PMF: \(p(x)=P(X=x)\) for discrete \(X\).
PDF: \(f(x)\ge 0\) for continuous \(X\); probabilities come from areas.
CDF: \(F(x)=P(X\le x)\).
Expected value: \(\mathbb{E}[X]\), the long-run average.
Variance: \(\mathrm{Var}(X)\), spread around the mean; \(\sigma=\sqrt{\mathrm{Var}(X)}\).
Binomial distribution: counts successes in \(n\) independent trials with success probability \(p\).
Normal distribution: bell-shaped distribution \(\mathcal{N}(\mu,\sigma^2)\); use z-scores to standardize.
Quick pre-check
Pre-check 1: Which random variable is continuous?
Hint: Continuous variables can take any value on an interval (including decimals).
Pre-check 2: For a standard normal random variable \(Z\), what is \(P(Z<0)\)?
Hint: The standard normal curve is symmetric about \(0\).
Distribution Basics
Random variables, PMF vs. PDF, and the CDF
Learning goal: Know when to sum (discrete) and when to integrate (continuous), and interpret probabilities as areas under a curve.
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: \[F(x)=P(X\le x).\]
Worked example
Example: In a continuous uniform distribution from \(0\) to \(12\), what is the probability a randomly chosen value is between \(4\) and \(8\)?
If \(X\sim\mathrm{Uniform}[0,12]\), then probabilities are proportional to interval length: \[P(4\le X\le 8)=\frac{8-4}{12-0}=\frac{4}{12}=\frac{1}{3}.\]
Try it
Try it 1: In a normal distribution, what is the probability that the variable is exactly equal to the mean?
Hint: A normal distribution is continuous, so any single point has probability \(0\).
Try it 2: What does the total area under the curve of a normal distribution represent?
Hint: For any PDF, the total area is \(1\).
Summary
Discrete: use a PMF and sum probabilities.
Continuous: use a PDF and integrate to get areas (probabilities).
Binomial Distribution
Bernoulli trials and the binomial distribution
Learning goal: Recognize binomial settings and compute probabilities with the binomial formula and the complement rule.
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: \[P(X=k)=\binom{n}{k}p^k(1-p)^{n-k},\quad k=0,1,\dots,n.\]
Worked example
Example: Let \(X\sim\mathrm{Bin}(n=2,p=0.5)\). What is \(P(X \ge 1)\)?
Use the complement rule: \[P(X\ge 1)=1-P(X=0).\] Compute \(P(X=0)\): \[P(X=0)=\binom{2}{0}(0.5)^0(0.5)^2=(0.5)^2=\frac{1}{4}.\] So: \[P(X\ge 1)=1-\frac{1}{4}=\frac{3}{4}.\]
Try it
Try it 1: A fair coin is flipped \(5\) times. Which expression gives the probability of getting exactly \(3\) heads?
Hint: Use \(\binom{n}{k}p^k(1-p)^{n-k}\) with \(n=5\), \(k=3\), \(p=\tfrac12\).
Try it 2: Which information must be known to use the binomial formula?
Hint: Binomial probabilities depend on the number of trials \(n\), success probability \(p\), and the target count \(k\).
Summary
Binomial model: \(X\sim\mathrm{Bin}(n,p)\) for counting successes in independent Bernoulli trials.
Complement rule is fast for "at least one": \(P(X\ge 1)=1-P(X=0)\).
Binomial Mean & Variance
Mean, variance, and quick binomial probability shortcuts
Learning goal: Use \(\mathbb{E}[X]\), \(\mathrm{Var}(X)\), and simple shortcut probabilities like \(P(X=0)\) and "at least zero."
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 \(0\).
Worked example
Example: For \(X\sim\mathrm{Bin}(n=5,p=0.5)\), what is \(P(X=0)\)?
Zero successes means every trial is a failure: \[P(X=0)=(1-0.5)^5=(0.5)^5=\frac{1}{32}.\]
Try it
Try it 1: Let \(X\sim\mathrm{Bin}(10,0.3)\). What is \(\mathrm{Var}(X)\)?
Hint: \(\mathrm{Var}(X)=np(1-p)=10(0.3)(0.7)\).
Try it 2: What is the probability of getting at least zero successes in a binomial distribution?
Hint: A binomial count \(X\) can never be negative, so \(X\ge 0\) always happens.
Summary
Binomial mean: \(\mathbb{E}[X]=np\).
Binomial variance: \(\mathrm{Var}(X)=np(1-p)\).
Quick facts: \(P(X=0)=(1-p)^n\) and \(P(X\ge 0)=1\).
Uniform Distribution
Continuous uniform distribution on \([a,b]\)
Learning goal: Compute uniform probabilities quickly using interval lengths, and know the key mean/variance formulas.
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: \(\mathrm{Var}(X)=\dfrac{(b-a)^2}{12}\)
Worked example
Example: In a continuous uniform distribution on \([0,10]\), what is the probability that a random value lies between \(2\) and \(4\)?
Use the length ratio: \[P(2\le X\le 4)=\frac{4-2}{10-0}=\frac{2}{10}=0.2.\]
Try it
Try it 1: If a value is picked at random from \([a,b]\), what is the probability it is in the first half of the interval?
Hint: Each half has the same length, so each half has probability \(\tfrac12\).
Try it 2: Let \(X\sim\mathrm{Uniform}[2,8]\). What is \(\mathrm{Var}(X)\)?
Normal distribution: symmetry, mean, and standard deviation
Learning goal: Interpret the normal curve correctly: probabilities are areas, and symmetry gives fast answers about the mean.
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 is continuous: \(P(X=\mu)=0\).
Worked example
Example: In a normal distribution, what is the probability that a value is less than the mean?
The normal curve is symmetric about the mean \(\mu\), so exactly half the area lies to the left: \[P(X<\mu)=\frac{1}{2}.\]
Try it
Try it 1: In a normal distribution, what is the probability that a value is above the mean?
Hint: Symmetry about \(\mu\) splits the area into two equal halves.
Try it 2: If the standard deviation of a normal distribution increases, what happens to the shape of the curve?
Hint: Larger \(\sigma\) spreads values out more, so the peak must lower to keep total area \(1\).
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 \(1\)).
Z-Scores
Standard normal distribution and z-scores
Learning goal: Convert normal values into z-scores so you can use standard normal probabilities.
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.
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.
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the distribution skill you need.
Practice set
Distribuzioni discrete e continue I practice questions with instant score
Answer all 10 questions below, then get your final score and a mistake review at the end so you know exactly what to improve.
0/10answered
Question 1Not answered
Una moneta equa viene lanciata \(5\) volte. Quale espressione dà la probabilità di ottenere esattamente \(3\) teste?
Correct answer: C. \(\binom{5}{3}\,0.5^{3}\,0.5^{2}\)
Explanation: Con la formula binomiale, \(P(X=k)=\binom{n}{k}p^{k}(1-p)^{n-k}\). Qui \(n=5, k=3, p=0.5\), quindi \(P(X=3)=\binom{5}{3}(0.5)^3(0.5)^2 = 10 \times 0.125 \times 0.25 = 0.3125\).
Question 2Not answered
Quale affermazione sulla distribuzione normale è vera?
Correct answer: C. La media, la mediana e la moda sono uguali
Explanation: La curva normale è simmetrica rispetto alla sua media; quindi media, mediana e moda coincidono.
Question 3Not answered
Quale delle seguenti situazioni \(\textbf{non}\) è descritta da una distribuzione binomiale?
Correct answer: C. Misurare le altezze di \(30\) studenti
Explanation: Le altezze sono misure continue; i modelli binomiali descrivono solo conteggi discreti di successi/insuccessi.
Question 4Not answered
Quale formula rappresenta la probabilità di esattamente \(k\) successi in \(n\) prove indipendenti (probabilità di successo \(p\))?
Correct answer: B. \(\binom{n}{k}p^{k}(1-p)^{n-k}\)
