Common Distributions — Hard Practice Quiz

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

Practice quiz

1. A Binomial distribution has $n$ trials and a probability of success $p$. When this distribution is approximated by a Normal distribution, what are the mean $\mu$ and standard deviation $\sigma$ of the approximating Normal distribution?

   • $\mu = np$, $\sigma = \sqrt{np(1-p)}$
   • $\mu = np(1-p)$, $\sigma = np$
   • $\mu = n(1-p)$, $\sigma = \sqrt{np}$
   • $\mu = p$, $\sigma = \sqrt{n(1-p)}$

   Answer: $\mu = np$, $\sigma = \sqrt{np(1-p)}$

2. For a Binomial distribution with a very large number of trials $n$ and a very small probability of success $p$, such that $np$ is a finite, moderate value, it can be approximated by a Poisson distribution. What is the parameter $\lambda$ of this approximating Poisson distribution?

   • $\lambda = p$
   • $\lambda = n$
   • $\lambda = np$
   • $\lambda = np(1-p)$

   Answer: $\lambda = np$

3. A binomial experiment consists of $100$ trials, and the probability of success in each trial is $0.2$. If this binomial distribution is approximated by a Normal distribution, what are the mean and variance of the approximating Normal distribution?

   • Mean $= 20$, Variance $= 16$
   • Mean $= 20$, Variance $= 4$
   • Mean $= 16$, Variance $= 20$
   • Mean $= 100$, Variance $= 20$

   Answer: Mean $= 20$, Variance $= 16$

4. A Poisson process has an average rate of $36$ events per unit time. If this Poisson distribution is approximated by a Normal distribution, what are the mean and standard deviation of the approximating Normal distribution?

   • Mean $= 36$, Standard Deviation $= 6$
   • Mean $= 36$, Standard Deviation $= 36$
   • Mean $= 6$, Standard Deviation $= 36$
   • Mean $= 18$, Standard Deviation $= 6$

   Answer: Mean $= 36$, Standard Deviation $= 6$

5. For a Binomial distribution with parameters $n$ and $p$, which of the following conditions is generally considered sufficient for a good Normal approximation?

   • $n$ is large, and $p$ is close to $0.5$.
   • $np < 5$ and $n(1-p) < 5$.
   • $np \ge 5$ and $n(1-p) \ge 5$.
   • $n$ is small, and $p$ is very small.

   Answer: $np \ge 5$ and $n(1-p) \ge 5$.

6. For a Normal distribution with probability density function $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$, at what value of $x$ is $f(x)$ maximized, and what is the maximum value?

   • $x = \mu$, maximum value is $\frac{1}{\sigma\sqrt{2\pi}}$
   • $x = \sigma$, maximum value is $\frac{1}{\mu\sqrt{2\pi}}$
   • $x = 0$, maximum value is $\frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{\mu^2}{2\sigma^2}}$
   • $x = \mu$, maximum value is $1$

   Answer: $x = \mu$, maximum value is $\frac{1}{\sigma\sqrt{2\pi}}$

7. A Binomial distribution has a mean $E(X) = 12$ and a variance $\text{Var}(X) = 4.8$. Determine the number of trials $n$ and the probability of success $p$.

   • $n = 20$, $p = 0.6$
   • $n = 12$, $p = 0.4$
   • $n = 30$, $p = 0.4$
   • $n = 25$, $p = 0.48$

   Answer: $n = 20$, $p = 0.6$

8. If a Poisson distribution has a variance of $4$, what is the probability of observing exactly $2$ events?

   • $8e^{-4}$
   • $4e^{-4}$
   • $16e^{-4}$
   • $2e^{-4}$

   Answer: $8e^{-4}$

9. For a standard Normal distribution (mean $\mu=0$, standard deviation $\sigma=1$), what is the ratio of the probability density at $x=2$ to the probability density at $x=0$?

   • $e^{-2}$
   • $e^{-1}$
   • $e^{-4}$
   • $1$

   Answer: $e^{-2}$

10. Consider a scenario where the number of events $X$ follows a Binomial distribution with parameters $n$ and $p$. If $n$ is very large and $p$ is very small, such that $np$ is a moderate constant, which of the following statements is true regarding approximations?

    • The Binomial distribution can be approximated by a Normal distribution with mean $np$ and variance $np(1-p)$.
    • The Binomial distribution can be approximated by a Poisson distribution with parameter $\lambda = np$.
    • The Poisson distribution (from B) can then be approximated by a Normal distribution if $\lambda$ is large.
    • All of the above.

    Answer: All of the above.