Random Variables

• A population is the entire set of things under study.

• E.g., all people in the world, all possible people in the world, all the neurons in the universe, the set of all possible action potential times relative to stimulus onset etc.

• The set of all possible outcomes of a random variable.

• E.g., the set of all possible rolls of a die is $\{1, 2, 3, 4, 5, 6\}$. It is discrete and finite.

• E.g., the set of all possible action potential times relative to stimulus onset spans the interval $(0, \infty)$. It is continuous and infinite.

• Describes the likelihood of each outcome in the sample space.

• Discrete: A function that gives the probability of each outcome.

• Continuous: A function that gives the probability density for each outcome.

• The sample space goes on the x-axis.

• The probability of each outcome goes on the y-axis.

• For a discrete random variable, the probability mass function (PMF) gives the probability of each outcome.

• Example: For a biased coin, the PMF is

  $$P(X = x) = \begin{cases} 0.3 & \text{if } x = \text{heads} \\ 0.7 & \text{if } x = \text{tails} \end{cases}$$

• Example: For a fair six-sided die, the PMF is $P(X = x) = \frac{1}{6}$ for $x \in \{1, 2, 3, 4, 5, 6\}$.

• Example: flipping a coin three times

• For a continuous random variable, the probability density function (PDF) gives the probability density for each outcome.

• Example: Distribution of heights in Australia

• Sample space: $(0, \infty)$

• Example: For a normal distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$, the PDF is

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

• Discrete case: The probability of a particular outcome $x$ is $P(X = x)$.

• Read this from a probability table or plot.

• Example: Probability of 7 Heads in 10 Flips

• Continuous case: The probability of a range of values is the area under the curve.

• Example: The probability of reaction time between 400-600 ms.

• Example: Probability of Reaction Time between 400-600 ms

• Consider a biased coin with PMF $P(X = x) = 0.3$ for heads and $P(X = x) = 0.7$ for tails.

• Samples obtained from this distrubtion are random sequences of heads and tails.

• Because the coin is biased, we expect to see more tails than heads.

A six-sided die is rolled, and the outcome is recorded. Which of the followin best describes the probability distribution of this random variable?

A. A probability density function (PDF) because the outcome is continuous.

B. A probability mass function (PMF) because the outcome is discrete.

C. A cumulative distribution function (CDF) because the outcome is deterministic.

D. A uniform density function because the outcome is continuous.

A researcher is studying the distribution of human reaction times in milliseconds (ms) to a visual stimulus. The reaction time is measured as a continuous variable. Which of the following best describes the probability distribution used to model this data?

A. A probability mass function (PMF) because reaction time is discrete.

B. A probability density function (PDF) because reaction time is continuous.

C. A cumulative distribution function (CDF) because reaction time is always increasing.

D. A binomial distribution because reaction time is categorical.

You are rolling a fair six-sided die and want to compute the probability of rolling a number greater than 4.

Which of the following correctly calculates this probability?

A. $P(X > 4) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$

B. $P(X > 4) = 1 - P(X \leq 4) = 1 - \frac{4}{6} = \frac{1}{3}$

C. $P(X > 4) = P(X = 5) \times P(X = 6) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$

D. $P(X > 4) = P(X = 4) + P(X = 5) + P(X = 6) = \frac{3}{6} = \frac{1}{2}$

A probability density function (PDF) is defined as follows:

• $f(x) = 0.1$ for $0 \leq x \leq 3$

• $f(x) = 0.2$ for $3 < x \leq 5$

• $f(x) = 0$ otherwise

• See next slide for the PDF plot.

• See next slide for quiz question.

What is the probability that a randomly drawn value from this distribution falls between $x=2$ and $x=5$?

A. $0.1 \times (3-2) + 0.2 \times (5-3) = 0.1 + 0.4 = 0.5$

B. $0.1 \times (5-2) = 0.3$

C. $0.2 \times (5-2) = 0.6$

D. $0.1 \times (3-2) + 0.2 \times (5-3) + 0.1 \times (5-4) = 0.1 + 0.4 + 0.1 = 0.6$

• See next slide for solution.