Statistics and Probabilities

What is the formula for the arithmetic mean of n data points

x1,x2\ldots x_{n}

\bar{x} = \frac{1}{n}\sum{i=1}^{n} xi [HB]

What measure of central tendency is resistant to extreme values?

The median — it depends only on the middle value(s), not on magnitude of outliers.

If a dataset has two modes, what is it called?

Bimodal.

Define variance for a population of N data points.

\sigma^2 = \frac{1}{N}\sum{i=1}^{N}(xi - \mu)^2 [HB]

Define sample variance.

s^2 = \frac{1}{n - 1}\sum{i=1}^{n}(xi - \bar{x})^2 — uses n-1

for an unbiased estimate. [HB]

What is standard deviation?

The square root of variance; \sigma = \sqrt{\frac{1}{N}\sum(x_i - \mu)^2} [HB]

What does a larger standard deviation indicate?

Greater data spread around the mean.

Define the range of a dataset.

The difference between the maximum and minimum values.

What is the coefficient of variation (CV)?

CV = \frac{\sigma}{\mu} \times 100\% — used to compare relative variability. [HB]

What is the probability of an event not occurring?

P(A') = 1 - P(A) [HB]

If two events A and B are independent, what is P(A \cap B)?

P(A)P(B) — product rule for independent events. [HB]

If two events are mutually exclusive, what is P(A \cap B)?

Zero — mutually exclusive events cannot occur together.

State the addition rule of probability.

P(A \cup B) = P(A) + P(B) - P(A \cap B) [HB]

What is conditional probability?

P(A|B) = \frac{P(A \cap B)}{P(B)} — probability of A given B has occurred. [HB]

State Bayes’ theorem.

P(A|B) = \frac{P(B|A)P(A)}{P(B)} — used to update probabilities with new information. [HB]

If P(A) = 0.4, P(B) = 0.3, P(A \cap B) = 0.12, find P(A \cup B).

P(A \cup B) = 0.4 + 0.3 - 0.12 = 0.58.

What are the two conditions for a discrete probability distribution?

(1) 0 \le P(x_i) \le 1 for all i

(2) \sum P(x_i) = 1.

For a fair six-sided die, what is P(X \le 4)?

\frac{4}{6} = \frac{2}{3}.

What is the expected value of a discrete random variable?

E[X] = \sum xi P(xi) [HB]

What is the variance of a discrete random variable?

Var(X) = \sum (xi - E[X])^2 P(xi) [HB]

What is the formula for the binomial probability distribution?

P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} [HB]

In a binomial distribution, what are the mean and variance?

\mu = np, \quad \sigma^2 = np(1 - p) [HB]

Give an example of a binomial distribution situation.

Number of heads in 10 coin flips.

What is the Poisson probability formula?

P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!} [HB]

When is the Poisson distribution used?

For the number of events in a fixed interval when events occur independently with a constant rate.

What are the mean and variance of a Poisson distribution?

Both equal to \lambda. [HB]

What is the probability density function (pdf) for a continuous uniform distribution?

f(x) = \frac{1}{b - a} for a \le x \le b. [HB]

What is the expected value of a continuous uniform distribution?

E[X] = \frac{a + b}{2} [HB]

What is the pdf of an exponential distribution?

f(x) = \lambda e^{-\lambda x}, \, x \ge 0 [HB]

What is the mean of an exponential distribution?

E[X] = \frac{1}{\lambda} [HB]

What is the cumulative distribution function (CDF) for an exponential distribution?

F(x) = 1 - e^{-\lambda x} [HB]

State the normal distribution probability density function.

f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} [HB]

What is the standard normal variable z?

z = \frac{x - \mu}{\sigma} [HB]

What percentage of data lies within 1, 2, and 3 standard deviations in a normal distribution?

68%, 95%, and 99.7% respectively — Empirical Rule. [HB]

If z = 1.96, what percentile does that correspond to?

About the 97.5th percentile — common in 95% confidence intervals.

What is a random sample?

A subset of a population selected such that each member has an equal chance of selection.

Define population vs. sample.

Population is the entire group; sample is a subset used to infer population characteristics.

What is a point estimate?

A single value used to estimate a population parameter (e.g., sample mean estimates population mean).

Define standard error of the mean.

SE = \frac{\sigma}{\sqrt{n}} — measures spread of sample means. [HB]

What is a 95% confidence interval for the mean when σ is known?

\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} [HB]

How is a 95% confidence interval interpreted?

We are 95% confident that the true population mean lies within the calculated interval.

State the null and alternative hypotheses in hypothesis testing.

H_0: no effect/difference; H_a: effect or difference exists.

When do you reject the null hypothesis?

When the p-value < α (significance level).

Define Type I and Type II errors.

Type I: reject true H_0; Type II: fail to reject false H_0.

What is the correlation coefficient r formula?

r = \frac{\sum (xi - \bar{x})(yi - \bar{y})}{\sqrt{\sum (xi - \bar{x})^2 \sum (yi - \bar{y})^2}} [HB]

Interpret r = 0.9.

Strong positive linear relationship between x and y.

What is the coefficient of determination r^2?

The proportion of variation in y explained by x — r^2 = 0.81 means 81% of variation is explained. [HB]

What are permutations and combinations used for?

Permutations: order matters P(n, r) = \frac{n!}{(n-r)!}; Combinations: order doesn’t C(n, r) = \frac{n!}{r!(n-r)!} [HB]
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