Discrete and Continuous Random Variables: Show A variable is a quantity whose value changes. A discrete variable is a variable whose value is obtained by counting. Examples: number of students present number of red marbles in a jar number of heads when flipping three coins students� grade level A continuous variable is a variable whose value is obtained by measuring. Examples: height of students in class weight of students in class time it takes to get to school distance traveled between classes A random variable is a variable whose value is a numerical outcome of a random phenomenon. ▪ A random variable is denoted with a capital letter ▪ The probability distribution of a random variable X tells what the possible values of X are and how probabilities are assigned to those values ▪ A random variable can be discrete or continuous A discrete random variable X has a countable number of possible values. Example: Let X represent the sum of two dice. Then the probability distribution of X is as follows:
To graph the probability distribution of a discrete random variable, construct a probability histogram. A continuous random variable X takes all values in a given interval of numbers. ▪ The probability distribution of a continuous random variable is shown by a density curve. ▪ The probability that X is between an interval of numbers is the area under the density curve between the interval endpoints ▪ The probability that a continuous random variable X is exactly equal to a number is zero Means and Variances of Random Variables: The mean of a discrete random variable, X, is its weighted average. Each value of X is weighted by its probability. To find the mean of X, multiply each value of X by its probability, then add all the products.
The mean of a random variable X is called the expected value of X. Law of Large Numbers: As the number of observations increases, the mean of the observed values,
The more variation in the outcomes, the more trials are needed to ensure that
Rules for Means: If X is a random variable and a and b are fixed numbers, then
If X and Y are random variables, then Example: Suppose the equation Y = 20 + 100X converts a PSAT math score, X, into an SAT math score, Y. Suppose the average PSAT math score is 48. What is the average SAT math score? Example: Let Let
The Variance of a Discrete Random Variable: If X is a discrete random variable with mean The standard deviation
Rules for Variances: If X is a random variable and a and b are fixed numbers, then
If X and Y are independent random variables, then
Example: Suppose the equation Y = 20 + 100X converts a PSAT math score, X, into an SAT math score, Y. Suppose the standard deviation for the PSAT math score is 1.5 points. What is the standard deviation for the SAT math score? Suppose the standard deviation for the SAT math score is 150 points, and the standard deviation for the SAT verbal score is 165 points. What is the standard deviation for the combined SAT score? *** Because the SAT math score and SAT verbal score are not independent, the rule for adding variances does not apply! What is the probability of tossing two heads if two coins are tossed?∴ The probability of getting exactly two heads is 1/4.
What is the possible value of a random variable if two coins are tossed?Consider a probability distribution in which the outcomes of a random event are not equally likely to happen. If the random variable Y is the number of heads we get from tossing two coins, then Y could be 0, 1, or 2. This means that we could have no heads, one head, or both heads on a two-coin toss.
What is the probability of exactly no heads in tossing a coin two times?(iv) getting no head:
Let E4 = event of getting no head. Then, E4 = {TT} and, therefore, n(E4) = 1. Therefore, P(getting no head) = P(E4) = n(E4)/n(S) = ¼.
|
If X is the random variable assigned to no of heads in tossing two coins then the variance is
zusammenhängende Posts
Urheberrechte © © 2024 ihoctot Inc.
