GATE Ecology and Evolution Mathematics and Quantitative Ecology Syllabus

What is the formula for combinations of $n$ objects taken $r$ at a time?

$$^{n}C_{r} = \binom{n}{r} = \frac{n!}{r!\,(n-r)!}$$ Order does not matter in combinations.

What is the key conceptual difference between a permutation and a combination?

In a permutation the order of selection matters (arrangements), whereas in a combination order does not matter (selections/groups). Thus $^{n}P_{r} = r! \cdot {}^{n}C_{r}$.

How is the classical (theoretical) probability of an event defined?

$$P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of equally likely outcomes}}$$

What is the range of any probability value, and what do $P=0$ and $P=1$ mean?

For any event, $0 \leq P(E) \leq 1$. $P(E) = 0$ means the event is impossible; $P(E) = 1$ means it is certain.

State the complement rule for probability.

$$P(E^{c}) = 1 - P(E)$$ where $E^{c}$ is the event that $E$ does not occur.

State the general addition rule for the probability of $A$ or $B$.

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ For mutually exclusive events, $P(A \cap B) = 0$, so $P(A \cup B) = P(A) + P(B)$.

State the multiplication rule for independent events.

If $A$ and $B$ are independent, then $$P(A \cap B) = P(A) \cdot P(B).$$

What does it mean for two events to be independent?

Two events are independent if the occurrence of one does not affect the probability of the other; formally $P(A \mid B) = P(A)$, equivalently $P(A \cap B) = P(A)P(B)$.

Define conditional probability $P(A \mid B)$.

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0$$ the probability of $A$ given that $B$ has occurred.

What distinguishes mutually exclusive events from independent events?

Mutually exclusive events cannot occur together ($P(A \cap B) = 0$). Independent events can occur together, with $P(A \cap B) = P(A)P(B)$. Two events with nonzero probability cannot be both mutually exclusive and independent.

What is a frequency distribution?

A frequency distribution is a table or summary showing how often each value or class interval of a variable occurs in a dataset, displaying the counts (frequencies) across categories.

Define relative frequency.

Relative frequency is the proportion of observations in a class: $$\text{relative frequency} = \frac{\text{class frequency}}{\text{total number of observations}}$$

What are the three common measures of central tendency?

Mean (arithmetic average), median (middle value when ordered), and mode (most frequently occurring value).

State the formula for the arithmetic mean of a sample.

$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_{i}$$

How are the median and mode defined?

The median is the middle value of an ordered dataset (average of the two middle values if $n$ is even). The mode is the value that occurs most frequently.

State the formula for the population variance $\sigma^{2}$.

$$\sigma^{2} = \frac{1}{N}\sum_{i=1}^{N} (x_{i} - \mu)^{2}$$ the mean of the squared deviations from the mean $\mu$.

State the formula for the sample variance $s^{2}$.

$$s^{2} = \frac{1}{n-1}\sum_{i=1}^{n} (x_{i} - \bar{x})^{2}$$ The divisor $n-1$ (Bessel's correction) gives an unbiased estimate.

What is the standard deviation and how does it relate to variance?

The standard deviation is the square root of the variance: $\sigma = \sqrt{\sigma^{2}}$ (or $s = \sqrt{s^{2}}$). It measures spread in the same units as the data.

What is the coefficient of variation (CV) and its formula?

The coefficient of variation is a relative measure of dispersion: $$CV = \frac{\sigma}{\mu} \times 100\%$$ It expresses the standard deviation as a percentage of the mean, allowing comparison of variability between datasets with different units or means.

Why is the coefficient of variation useful compared to the standard deviation alone?

Because it is unitless (a ratio), the CV allows comparison of variability across datasets that have different units or very different means, which raw standard deviations cannot do fairly.

What does the correlation coefficient measure, and what is its range?

The (Pearson) correlation coefficient $r$ measures the strength and direction of a linear relationship between two variables. It ranges $-1 \leq r \leq 1$.

Interpret the values $r = 1$, $r = -1$, and $r = 0$.

$r = 1$: perfect positive linear correlation; $r = -1$: perfect negative linear correlation; $r = 0$: no linear correlation between the variables.

State the formula for the Pearson correlation coefficient.

$$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^{2}\,\sum (y_i - \bar{y})^{2}}}$$

Why does correlation not imply causation?

A statistical correlation between two variables only indicates they vary together; it does not prove one causes the other. The association may be due to a confounding (lurking) variable, reverse causation, or coincidence.

What is a p-value?

A p-value is the probability of obtaining a test result at least as extreme as the observed one, assuming the null hypothesis $H_0$ is true. A small p-value indicates the data are unlikely under $H_0$.

How is a p-value used to make a decision at significance level $\alpha$?

If $p \leq \alpha$ (e.g. $\alpha = 0.05$), reject the null hypothesis $H_0$ (result is statistically significant). If $p > \alpha$, fail to reject $H_0$.

Define a Type I error and its probability.

A Type I error is rejecting the null hypothesis $H_0$ when it is actually true (a false positive). Its probability equals the significance level $\alpha$.

Define a Type II error and its probability.

A Type II error is failing to reject the null hypothesis $H_0$ when it is actually false (a false negative). Its probability is denoted $\beta$, and statistical power $= 1 - \beta$.

Compare Type I and Type II errors in a hypothesis test.

Type I error ($\alpha$): rejecting a true $H_0$ (false positive). Type II error ($\beta$): failing to reject a false $H_0$ (false negative). Decreasing $\alpha$ generally increases $\beta$, all else equal.

What is a test statistic in hypothesis testing?

A test statistic is a value computed from sample data (e.g. a $z$, $t$, $\chi^{2}$, or $F$ statistic) that measures how far the sample result deviates from what is expected under $H_0$. It is compared to a reference distribution to obtain the p-value or critical value for the decision.