Common Distributions — Practice Quiz

A Probability cheat sheet for Common Distributions — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Binomial Distribution

$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

Where: \(n\) = trials, \(p\) = probability of success

Binomial Mean and Variance

$$E(X) = np, \quad \text{Var}(X) = np(1-p)$$

Poisson Distribution

$$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$$

Where: \(\lambda\) = average rate

Normal Distribution PDF

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

Where: \(\mu\) = mean, \(\sigma\) = std dev

Practice quiz

  1. A fair coin is tossed $10$ times. What is the probability of getting exactly $7$ heads?

    • $\frac{15}{128}$
    • $\frac{10}{128}$
    • $\frac{21}{256}$
    • $\frac{3}{32}$

    Answer: $\frac{15}{128}$

  2. In a binomial experiment with $n=20$ trials and a probability of success $p=0.4$, what are the mean and variance of the number of successes?

    • Mean $= 8$, Variance $= 4.8$
    • Mean $= 8$, Variance $= 8$
    • Mean $= 4.8$, Variance $= 8$
    • Mean $= 4.8$, Variance $= 4.8$

    Answer: Mean $= 8$, Variance $= 4.8$

  3. The average number of calls received by a call center in an hour is $5$. What is the probability that the call center receives exactly $3$ calls in the next hour, assuming a Poisson distribution?

    • $\frac{125 e^{-5}}{6}$
    • $\frac{25 e^{-5}}{6}$
    • $\frac{125 e^{-3}}{6}$
    • $\frac{5 e^{-5}}{3}$

    Answer: $\frac{125 e^{-5}}{6}$

  4. For a normal distribution with mean $\mu$ and standard deviation $\sigma$, what does the parameter $\sigma$ primarily control in the probability density function $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$?

    • The center of the distribution
    • The skewness of the distribution
    • The spread or width of the distribution
    • The height of the peak of the distribution

    Answer: The spread or width of the distribution

  5. Which of the following scenarios is best modeled by a Poisson distribution?

    • The number of heads obtained in $100$ coin flips.
    • The number of defective items in a sample of $50$ from a large production batch.
    • The number of earthquakes of magnitude $5.0$ or higher occurring in a specific region per year.
    • The height of adult males in a country.

    Answer: The number of earthquakes of magnitude $5.0$ or higher occurring in a specific region per year.

  6. A factory produces light bulbs, and $5\%$ of them are defective. If a random sample of $10$ bulbs is taken, what is the expected number of defective bulbs in the sample?

    • $0.05$
    • $0.5$
    • $1$
    • $5$

    Answer: $0.5$

  7. A website experiences an average of $2$ server crashes per month. Assuming a Poisson distribution, what is the probability of having no server crashes in a given month?

    • $e^{-2}$
    • $2e^{-2}$
    • $\frac{e^{-2}}{2}$
    • $1 - e^{-2}$

    Answer: $e^{-2}$

  8. In the normal distribution probability density function $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$, what does the parameter $\mu$ represent?

    • The standard deviation of the distribution
    • The variance of the distribution
    • The mean of the distribution
    • The probability of success

    Answer: The mean of the distribution

  9. A basketball player has a $70\%$ free throw success rate. If they attempt $50$ free throws, what is the variance of the number of successful free throws?

    • $10.5$
    • $35$
    • $15.75$
    • $24.5$

    Answer: $10.5$

  10. Which of the following probability distributions is characterized by the property that its mean is equal to its variance?

    • Binomial distribution
    • Normal distribution
    • Poisson distribution
    • Uniform distribution

    Answer: Poisson distribution

Select a subject

Select a subject from the left panel to begin exploring formulas.