STAT 302 Course Notes set 4 1 Random Variables In set 2 notes we dealt with variables. In set 3 of the notes we calculated simple probabilities such as: the probability a randomly chosen 60-year-old women gets breast cancer. Combining the ideas of set 2 and set 3 notes: The Random Variable Random variable: - A random variable is a numerical measurement of the outcome of a random phenomenon. - We usually let X stand for the random variable. Probability distribution of a random variable: - The probability distribution of a random variable is telling us what values the random variable takes and how to assign probabilities to those values. Terminology and Notation: - Capital letters, such as X, are used to represent the random variable. - Lower case letters, like x, refer to particular values taken by the random variable. Discussion of a typical situation: Consider the population of all babies born between 25 and 28 weeks of age. The birth weight is the measurement of interest. Birth weight is a variable as discussed in sets 1 and 2 of the notes. Scenario 1: A baby is chosen at random (which makes this a random phenomenon) and the birth weight is measured.  The random variable X is the birth weight of a baby randomly chosen from this population.  The measurement is numerical and the value of the random variable the weight.  The probability distribution is based on the distribution of the variable birth weights.  P(X < 900 gr) = probability a randomly choosing a baby who weighs less than 900 grams at birth = proportion of all babies in this population whose birth weight is less than 900 grams. Scenario 2: The variable of interest is categorical taking 2 values (i.e. get cancer, doesn’t get cancer). In this scenario we randomly select n subjects and count the number of “successes” such as counting the number of subject who don’t get cancer. The count is the variable value. This is the scenario associated with the binomial random variable and is the topic we cover at the end this set.STAT 302 Course Notes set 4 2 Scenario 3: Now suppose 30 babies are chosen at random from the population. The random variable X is the average birth weight of the 30 babies selected at random. Now the probability distribution of the random variable is quite different from the distribution of the variable = birth weights. Scenario 3 is the topic of set 5. Discrete random variable: - A discrete random variable takes a set of separate values such as 0, 1, 2, 3, … For discrete random variables, you can list all possible outcomes along with their associated probabilities. You can find the probability for any event by adding the probabilities of the individual outcomes that make up the event. EX 1: Pure dog breeds are often highly inbred, leading to high numbers of congenital defects. A study examined hearing impairment in 5333 Dalmatians.  Let X = the number of ears impaired (deaf) in a randomly chosen Dalmatian Probability distribution of X: Find the following probabilities: - P(X = 1) Express P(X=1) in words: - P(X ≥ 1) - P(X > 1) Continuous random variables: - The possible values taken by a continuous variable form an interval. Consequently, there are an infinite number of possible values for a continuous random variable. X = weight of randomly chosen subject in a population. The possible values taken by this randomSTAT 302 Course Notes set 4 3 variable is the interval [weight of least heavy subject, weight of heaviest] The shape of this distribution is bell shaped. - Because there are an infinite number of possible outcomes, it is impossible to find probabilities by adding as we did with discrete random variables. - Instead, we will calculate probabilities by finding areas under a density curve. X = weight of a randomly chosen subject from the population. The density curve would look like a smoothed out histogram of the weights of all people in the population but with a different scale for the vertical axis. Density curves: - The density curve describes the overall pattern of a distribution. The area defined by this curve is 1. - You can think of a density curve as a smooth line drawn over the tops of the bars in a narrow bin histogram. The y-axis (vertical) scale has to be changed to make the total area of the bars equal to 1. We calculate probabilities by finding areas under the density curve Boardwork:STAT 302 Course Notes set 4 4 Boardwork: EX: X = distance, in feet, a randomly selected 5 year old can throw a baseball. The probability a randomly selected child can throw between 3 and 5 feet is the proportion of children in the population who can throw a baseball between 3 and 5 feet. Suppose we have a census of the population. Then P( 3 < X < 5) = We can think of this in terms of areas as follows: P(3 < X < 5)area under density between 3 and 5 = = area under density curve between 3 and 5total area under density curve Boardwork:STAT 302 Course Notes set 4 5 Handy facts for continuous random variables: P(X = c) = 0 for any number c. P(X ≤ c) = P(X <