Chapter 5: Random Variables and Discrete Probability Distributions

What are the properties of a valid probability distribution?

1. Each probability has to be between 0 and 1 inclusively. In symbols, 0 ≤ p ≤ 1 2. The sum of all of the probabilities must equal 1. In symbols, ∑ ​pi​(x) = 1

What is a random variable (abbreviated as rv)?

A random variable (abbreviated as rv), is a numerical characteristic that is obtained from a random experiment. Therefore, random variables are functions and have to follow all properties of mathematical functions. This means that there can be only one number associated with each outcome.

What is the expected value of a random variable?

If we want to emphasize that μ (the mean) is for the variable x, we write μx. This is the average of all possible values of the random variable X. This is also called the expected value, E(X).

What does the I in BInS stand for?

Independent - The outcomes of the trials have to be independent of each other. That is, if we know the results of one trial, that does not affect the result of any other trial.

What is the Poisson distribution generally used for?

The Poisson distribution is used when looking at the number of counts per time or area.

When do we use the Poisson random variable?

The Poisson random variable is used when we are interested in another count. This time, it is the number of times that an event occurs during a particular time interval or in a particular area.

How do we describe the Poisson random variable?

The Poisson random variable only has one parameter, λ (lambda, the Greek 'el'). We say that X is distributed as a Poisson random variable with parameter lambda, X ~ Poisson( λ).

Some discrete distributions occur over and over again, and so formulas have been derived for them. What are the two common distributions that we will cover?

The binomial distribution and the Poisson distribution.

What is the cumulative distribution function (abbreviated as "cdf")?

The cdf is defined as the probability that P(X ≤ x). Note the equal sign in the probability.

If X ~ Poisson (λ), then which of the following are equal to λ?

The variance and the mean.

What is the binomial random variable referred to as?

We say that X is distributed as a binomial random variable with parameters n and p. This is written as X ~ B(n, p) where the ~ is read as 'is distributed as.'

What is the symbol for the mean of a random variable?

μ is the population mean so we will use this as the symbol for the mean of a random variable. If we want to emphasize that μ is for the variable x, we write μx.

(True/False) For X to be a binomial random variable, we have to have a binomial experiment (BInS occurs) and X is the number of successes.

True

What is a continuous random variable?

A continuous random variable can have all possible values in one or more intervals. Usually continuous random variables are things that are measured.

What is a discrete random variable?

A discrete random variable can only have a countable number of values, usually these values are positive or non-negative integers.

At is the acronym used to determine if an experiment is a binomial experiment?

BInS

What are random variables abbreviated by?

Capital Latin letters such as A, B, etc.

Rule 3 of variances applies if X and Y have correlation ρ. What is rule 3 of variances?

If X and Y are dependent on each other, there is quantity rho (ρ) which is called the correlation which changes the formula. Again, you add the variances, but then the sign of the correlation term depends on the original operator. Again, you always calculate the variance first and then take the square root to obtain the standard deviation.

What are the shapes of a binomial distribution?

If p is less than 0.5 then the distribution is right skewed. If p = 0.5, then the distribution is symmetrical. If p > 0.5, then the distribution will be left skewed. How skewed the distribution is depends on the both the value of n and the value of p if p ≠ 0.5

What is the binomial distribution generally used for?

It is generally used to model proportions.

What are some examples of Poisson experiments?

The following are examples of Poisson random variables: - X is the number of people who enter a particular location like the Union during a specific time period. - X is the number of a-particles emitted from a radioactive nuclide like Uranium-238 in a certain time period. - X is the number of DNA fragments found from a sequencing experiment. - X is the number of dead trees in a certain area of a forest.

How do you find the mean of a random variable given a probability distribution?

You take each possible value of X, multiply it by its probability, and then sum all of those products up.

What are the 3 parts to determine if an experiment is a Poisson experiment?

1. The probability that a particular event will occur in a given interval (of time, length, volume, etc.) is the same for all units of equal size and the rate is proportional to the size of the unit. This is the equivalent of the probability of a success being constant in the binomial experiment. 2. The number of events that occur in any interval is independent of the number that occur in any other non-overlapping interval. This is our assumption of independence. 3. The probability that more than one event occurs in a unit of measure is negligible for very small-sized units. In other words, the events only occur one at a time if Δt is very small.

What does the B in BInS stand for?

Binary - There are only two possible outcomes for each trial. By convention we call them success, S, and failure, F. The words 'success' and 'failure' are technical terms. A success is what we are interested in, a failure is what we are not interested in.

(True/False) The cumulative distribution function (cdf) is usually used for discrete distributions.

False, cdf is usually used for continuous distributions.

(True/False) To find the standard deviation of σ^2(X±Y) you can add the standard deviations of X and Y together.

False, the standard deviation is the square root of the variance not the sum of the standard deviations.

(True/False) The units of λ are always the same as the units provided in the question.

False, the units may be different. For example if the units of λ are per hour, but the question asks about something for two hours, λ' = 2λ. For this reason, I strongly suggest that you always include the units for λ.

(True/False) The variance can be negative.

False, the variance can never be negative.

Rule 2 for variance is called the addition rule for variances. What is rule 2 for variance?

No matter what the mathematical operator is between X and Y, you always add the two variances together. Remember if you want the standard deviation, you take the square root of the variance; it is not the sum of the two standard deviations.

What are probability distributions?

Random variables also have distributions. These are called probability distributions and let us know what all of the possible values are for each random variable. They also provide the probability that each outcome will occur.

Rule 1 for means is used when you want to calculate the mean of a linear function of X. What is rule 1 for means?

Remember that a linear function is a polynomial with no power of the variable greater than 1. That is, you can take the constants 'out' of the equation. This is called the linearity rule for means.

Rule 2 for means is used when you want to combine two different random variables X and Y. What is rule 2 of means?

Rule 2 is used when you want to combine two different random variables X and Y. Here, you can just take the variables 'out' of the equation also. This is called the addition rule for means and can be extended to any number of random variables. The only mathematical operators that you can use are + or -. Rules 1 and 2 are often combined.

Rule 3 of means is for functions of a random variable that are not linear. What is rule 3 of means?

Rule 3 is for functions of a random variable that are not linear. In this case, all you need to do is replace xi with g(xi) in the formula for the expected value. This should only be used when g(x) is NOT a linear function since Rule 1 which is for linear functions is much easier to use.

What does the S in BInS stand for?

Success - stands for the probability of a success being constant. This is written P(S) = p where p is the constant. There is a lot of confusion about what is abbreviated by 'p.' A capital P stands for probability. A small p stands for the probability of success. In addition, if the small p is a function, it stands for the pmf (probability mass function) of a random variable

What is the symbol for population variance?

The Greek letter for s is σ; therefore, the population variance is σ^2. This is also abbreviated by Var(X).

What is a binomial random variable?

The binomial random variable, X, has to be from a binomial experiment and maps each outcome in the binomial experiment to a real number. X is defined to be the number of successes. X has the parameters n and p where n is the number of trials and p is the probability of a success. If you know n and p, then you can calculate all of the probabilities for X.

What is the formula to calculate the number of combinations?

The combinations of "n choose x" is the number of times that we have x successes and n - x failures. This is called the binomial coefficient.

(True/False) For a Poisson distribution, the number of possibilities for x starts at 0 and is countably infinite. x can not be negative because it doesn't make sense to have negative counts.

True

(True/False) Random variables can either be discrete or continuous.

True

(True/False) Standard deviations are always calculated by taking the square root of the variance. They are never calculated by adding and/or subtracting the standard deviations.

True

(True/False) The parameter λ is the same as the mean.

True

(True/False) The possible values of X are from 0 to n, where n is the number of trials.

True

(True/False) The rule for a random variable is called a pmf.

True

(True/False) The skewness of a binomial random variable depends on the value of p.

True

What is the probability mass function or pmf of the random variable?

We call the probability distributions or rules of a discrete random variable, the probability mass function or pmf of the random variable. In symbols, p(x) = P(X= x). The pmf can be displayed as a table or a function.

What does the n in BInS stand for?

n - refers to the fact that the number of trials is fixed.