lecture 4 - Discrete Random Variables

What is a discrete random variable?

A variable that can take on a finite or countable number of values (e.g. number of calls per day).

What is a probability distribution function (PDF) for discrete variables?

It assigns probabilities to each value of a discrete random variable such that each probability is between 0 and 1, and they sum to 1.

What does a probability distribution table show?

All possible values of the random variable and their associated probabilities.

What is the expected value (mean) of a discrete random variable?

E(X) = sum of [x × P(x)] — it's the long-run average value.

What does E(X) represent?

It represents the central (average) value you'd expect if the experiment were repeated many times.

What is variance of a discrete random variable?

Var(X) = sum of [(x - mean)² × P(x)] or Var(X) = E(X²) - [E(X)]².

What is standard deviation of a discrete variable?

The square root of variance — shows average distance from the mean.

What is the formula for E(X²)?

E(X²) = sum of [x² × P(x)] — used to calculate variance.

What is a Bernoulli distribution?

A distribution with only two outcomes: success (1) and failure (0).

What are the mean and variance of a Bernoulli distribution?

Mean = p; Variance = p(1 - p), where p = probability of success.

What is a binomial distribution?

A distribution showing the number of successes in a fixed number of independent trials with the same success probability.

What are the conditions for a binomial experiment?

Fixed number of trials, each trial has 2 outcomes, constant p, and independent trials.

What is the binomial probability formula?

P(X = x) = C(n, x) × p^x × (1 - p)^(n - x), where C(n, x) = n! / [x!(n - x)!].

What are the mean and variance of a binomial distribution?

Mean = n × p; Variance = n × p × (1 - p).

What is a Poisson distribution?

A distribution used for counting events that happen over a fixed time or space with a known average rate.

What is the formula for Poisson probability?

P(X = x) = (e^(-λ) × λ^x) / x!, where λ is the average rate of events.

What are the mean and variance of a Poisson distribution?

Both the mean and variance equal λ.

When is the Poisson distribution appropriate?

When events happen randomly, independently, and at a constant average rate.

What is the difference between binomial and Poisson distributions?

Binomial is for a fixed number of trials; Poisson is for event counts over time/space.

What is λ (lambda) in the Poisson distribution?

It's the expected (average) number of occurrences in a fixed interval.