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Transcript
Chapter 16: Discrete and
Continuous Random Variables
 Variable
o Quantity whose value changes
 Discrete Variable
o Variable whose value is obtained by counting
 Number of students present
 Number of heads when flipping three coins
 Continuous Variable
o Variable whose value is obtained by measuring
 Height of students in class
 Time it takes to get to school
 Random Variable
o Variable whose value is a numerical outcome of a random
phenomenon
o Denoted with a capital letter
o Probability distribution of a random variable X tells what the possible
value of X are and how the probabilities are assigned to those values
o Random variable can be discrete or continuous
 Discrete Random Variable
o X has a countable number of possible values
o To graph the probability discrete random variable make a frequency
table
 Let X represent the sum of two dice
 Continuous Random Variable
o X takes all values in a given interval of numbers
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o The probability distribution of a continuous random variable is shown
by ******
o Probability that X is between an interval of numbers is the area under
the density curve between the interval endpoints
Means and Variances of Random Variables
o Mean of a discrete variable X is it’s weighted average
 Each value of x is weighted by its probability
o To find the mean of X
 Multiply each value of X by its probability
 Then add all the products
o The mean of a random variable X is called the ******* of X
Law of Large Numbers
o As the number of observations increase, the mean of observed
values x- bar approaches the mean of the population
 As sample size increases the sample mean gets closer to the
true mean (µ)
o The more variation in the outcomes, the more trials are needed to
ensure x- bar is close to mew (µ) or the true mean
 The wider the variation the more sample sizes it’s going to take
to get the sample mean to the true mean
Rules for Means
o If X is a random variable and a and b are fixed numbers
 𝜇𝑎+𝑏𝑋 = 𝑎 + 𝑏𝜇𝑥
o If X and Y are random variables
 𝜇𝑋+𝑌 = 𝜇𝑥 + 𝜇𝑌
Variance of a Discrete Random Variable
o If X is a discrete random variable with mean (µ)
o 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑋 (𝜎𝑋2 = ∑(𝑥𝑖 − 𝜇𝑥 )2 𝑃𝑖
o Standard deviation 𝜎𝑥 is the square root of the variance
Rules for Variances
o If X is a random variable and a and b are fixed numbers
2
𝜎𝑎+𝑏𝑋
= 𝑏 2 𝜎𝑋2
o If X and Y are independent random variables
2
 𝜎𝑋+𝑌
= 𝜎𝑋2 + 𝜎𝑌2
2
 𝜎𝑋−𝑌
= 𝜎𝑋2 + 𝜎𝑌2