4.7 - Discrete Random Variables

Front

What is a discrete random variable?

Back

A discrete random variable is a type of variable that can take on a countable number of distinct values, often representing counts or whole numbers.

Front

What is a probability distribution?

Back

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

Front

What are the properties of a valid probability distribution?

Back

1. All probabilities must be between 0 and 1. 2. The sum of all probabilities must equal 1.

Front

How do you calculate the expected value (mean) of a discrete random variable?

Back

The expected value is calculated by multiplying each possible value by its probability and then summing all these products.

Front

What is the formula for expected value (E(X))?

Back

E(X) = Σ [x * P(x)], where x is a value and P(x) is the probability of x.

Front

What does it mean to find the probability of at least k successes in a binomial distribution?

Back

It means calculating the probability of getting k or more successes in a fixed number of trials.

Front

What is the binomial probability formula?

Back

P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success.

Front

What is the significance of the mean in a probability distribution?

Back

The mean represents the average outcome you would expect if you repeated the experiment many times.

Front

How do you interpret a negative expected value in a gambling scenario?

Back

A negative expected value indicates that, on average, you would lose money if you played the game many times.

Front

What is the difference between a fair game and an unfair game in terms of expected value?

Back

A fair game has an expected value of zero, while an unfair game has a negative expected value for the player.