Midterm 2 Study Guide: Probability, Distributions, and Confidence Intervals

Probability and Random Variables

Basic Probability Concepts

Probability is the measure of the likelihood that an event will occur. In statistics, we use probability to quantify uncertainty and make inferences about populations based on samples.

• Sample Space (S): The set of all possible outcomes of a random experiment.

• Event (A): A subset of the sample space; a collection of outcomes.

• Probability of an Event: $P(A)$, where $0 \leq P(A) \leq 1$ and $P(S) = 1$.

• Complementary Events: $P(A^c) = 1 - P(A)$

• Union and Intersection: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Example: If the probability that a student is late is 0.2, then the probability that the student is not late is $1 - 0.2 = 0.8$.

Random Variables

A random variable is a function that assigns a real number to each outcome in the sample space of a random experiment.

• Discrete Random Variable: Takes on a countable number of possible values (e.g., number of students late to class).

• Continuous Random Variable: Takes on any value in an interval (e.g., time spent on homework).

Example: Let $X$ be the number of students who arrive late to an exam. $X$ is a discrete random variable.

Probability Distributions

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

• Parameters: $n$ (number of trials), $p$ (probability of success)

• Probability Mass Function (PMF): $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Mean: $E[X] = np$

• Variance: $Var(X) = np(1-p)$

Example: If 20 students each have a 0.3 probability of being late, the probability that exactly 5 are late is $P(X=5) = \binom{20}{5} (0.3)^5 (0.7)^{15}$.

Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval of time or space, given the events occur independently and at a constant average rate.

• Parameter: $\lambda$ (average rate of occurrence)

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

• Mean and Variance: $E[X] = Var(X) = \lambda$

Example: If the average number of emails received per hour is 5, the probability of receiving exactly 3 emails in an hour is $P(X=3) = \frac{e^{-5} 5^3}{3!}$.

Uniform Distribution

The uniform distribution is a continuous probability distribution where all intervals of the same length within the distribution's range are equally probable.

• PDF: $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$

• Mean: $E[X] = \frac{a+b}{2}$

• Variance: $Var(X) = \frac{(b-a)^2}{12}$

Example: If the time to grade an assignment is uniformly distributed between 1 and 5 days, the probability it takes more than 3 days is $P(X > 3) = \frac{5-3}{5-1} = 0.5$.

Exponential Distribution

The exponential distribution models the time between events in a Poisson process.

• Parameter: $\lambda$ (rate parameter)

• PDF: $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$

• Mean: $E[X] = \frac{1}{\lambda}$

• Variance: $Var(X) = \frac{1}{\lambda^2}$

Example: If the average time between emails is 2 hours ($\lambda = 0.5$), the probability that the next email arrives after 3 hours is $P(X > 3) = e^{-0.5 \times 3}$.

Sampling Distributions and Confidence Intervals

Sampling Distributions

A sampling distribution is the probability distribution of a given statistic based on a random sample.

• Central Limit Theorem (CLT): For large $n$, the sampling distribution of the sample mean is approximately normal, regardless of the population distribution.

• Standard Error (SE): $SE = \frac{\sigma}{\sqrt{n}}$

Example: If the population standard deviation is 2 and the sample size is 25, $SE = \frac{2}{5} = 0.4$.

Confidence Intervals for the Mean

A confidence interval gives a range of plausible values for a population parameter, such as the mean, based on sample data.

• Formula (known $\sigma$): $\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$

• Formula (unknown $\sigma$): $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

• Interpretation: A 90% confidence interval means that, in repeated sampling, 90% of such intervals would contain the true mean.

Example: If $\bar{x} = 3.5$, $s = 0.5$, $n = 16$, and $t^* = 1.753$ for 90% confidence, the interval is $3.5 \pm 1.753 \times \frac{0.5}{4} = (3.281, 3.719)$.

Properties of Expectation and Variance

Linearity of Expectation

The expected value operator is linear, meaning:

• $E[aX + bY] = aE[X] + bE[Y]$ for any random variables $X, Y$ and constants $a, b$.

Variance Properties

• $Var(aX + b) = a^2 Var(X)$

• If $X$ and $Y$ are independent, $Var(X + Y) = Var(X) + Var(Y)$

Useful Probability and Statistics Formulas

Concept	Formula
Binomial PMF	$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
Poisson PMF	$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$
Uniform PDF	$f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$
Exponential PDF	$f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$
Mean of Binomial	$E[X] = np$
Variance of Binomial	$Var(X) = np(1-p)$
Mean of Poisson	$E[X] = \lambda$
Variance of Poisson	$Var(X) = \lambda$
Mean of Uniform	$E[X] = \frac{a+b}{2}$
Variance of Uniform	$Var(X) = \frac{(b-a)^2}{12}$
Mean of Exponential	$E[X] = \frac{1}{\lambda}$
Variance of Exponential	$Var(X) = \frac{1}{\lambda^2}$
Standard Error	$SE = \frac{\sigma}{\sqrt{n}}$
Confidence Interval (mean, known $\sigma$)	$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$
Confidence Interval (mean, unknown $\sigma$)	$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

Normal and t-Distributions

Standard Normal Distribution

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Probabilities are found using standard normal tables (z-tables).

• Z-score: $z = \frac{x - \mu}{\sigma}$

t-Distribution

The t-distribution is used instead of the normal distribution when the population standard deviation is unknown and the sample size is small. It is similar to the normal distribution but has heavier tails.

• Degrees of Freedom: $df = n - 1$

• Use t-tables to find critical values for confidence intervals.

Summary Table: Key Distributions

Distribution	Type	Parameters	Mean	Variance
Binomial	Discrete	$n, p$	$np$	$np(1-p)$
Poisson	Discrete	$\lambda$	$\lambda$	$\lambda$
Uniform	Continuous	$a, b$	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$
Exponential	Continuous	$\lambda$	$\frac{1}{\lambda}$	$\frac{1}{\lambda^2}$
Normal	Continuous	$\mu, \sigma^2$	$\mu$	$\sigma^2$

Application: Interpreting Confidence Intervals

When you construct a confidence interval for a mean, you are estimating the range in which the true population mean lies with a certain level of confidence (e.g., 90%).

• Correct Interpretation: "We are 90% confident that the average number of hours per week and per course UCSD students spend outside of class working on materials is between 3.5166 and 3.5186 hours."

• Incorrect Interpretation: "There is a 90% probability that the true mean is in this interval." (The interval either contains the mean or it does not; the confidence refers to the method, not the specific interval.)

Practice Problems and Solutions

• Problem: What is the probability that a random variable $X$ uniformly distributed on [1,5] exceeds 3?

• Solution: $P(X > 3) = \frac{5-3}{5-1} = 0.5$

• Problem: If $T \sim Exp(0.5)$, what is $P(T > 2)$?

• Solution: $P(T > 2) = e^{-0.5 \times 2} = e^{-1} \approx 0.368$

Additional info: This study guide is based on a midterm exam covering probability, discrete and continuous distributions (binomial, Poisson, uniform, exponential), properties of expectation and variance, and confidence intervals for means. It includes both conceptual and computational aspects, as well as interpretation of results.