6.2: Discrete Probability Distributions Flashcards
What is a discrete probability distribution?
A discrete probability distribution is a function that gives the probability associated with each possible value a discrete random variable can assume.
What must a discrete probability distribution satisfy?
A discrete probability distribution must satisfy that the probabilities are non-negative and sum to one.
What are the properties of a discrete random variable’s distribution?
The probabilities for each value of the discrete random variable must be greater than or equal to zero and their sum must equal one.
How is the mean or expected value of a discrete random variable calculated?
The mean, or expected value, of a discrete random variable is calculated as
μx = Σ[x*p(x)]
summing over all possible values of x.
How can a discrete probability distribution help in decision-making?
It allows for calculating the probabilities of various outcomes, aiding in understanding the likelihood of different events.
How do you compute the expected value (mean) of a discrete random variable?
The expected value (mean) is computed as
μx = Σ[x*p(x)]
which is the sum of each value multiplied by its probability.
How does the expected value relate to actual observed values?
The expected value is the long-run average if the experiment is repeated many times, not necessarily the average of any particular sample.
What is the variance of a discrete random variable?
The variance, denoted as σx², is the average of the squared deviations from the mean, calculated as
σx² = Σ[(x - μx)²*p(x)].
How do you calculate the standard deviation of a discrete random variable?
The standard deviation, σx, is the positive square root of the variance: σx = √σx².
Why might relying solely on expected value be misleading in some decisions?
Because expected value does not account for risk or variability. A single event with a high loss potential may not be worth the risk despite a positive expected value.
What does the standard deviation tell us about a random variable?
The standard deviation gives a measure of the spread or dispersion of the probability distribution of a random variable.
In the context of insurance, why is understanding the variance and standard deviation important?
It helps in assessing the risk associated with the insurance policy, reflecting the volatility of the company’s profit.
What is Chebyshev’s Theorem?
Chebyshev’s Theorem states that for any random variable, the probability that the outcome will be within k standard deviations of the mean is at least 1-1/k².
How do you interpret the standard deviation in a practical context?
Standard deviation measures the amount of variability or spread in a set of data.
What is the probability that a random variable will be within 2 standard deviations of the mean according to Chebyshev’s Theorem?
At least 1 - (1/2²) = 3/4 or 75%.
