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Probability Density Function

     A probability density function (pdf) is used when you are dealing with continuous random variables. Remember, a continuous random variable is something like time where there is no distinct separation between one value and the next as opposed to discrete random variables such as number of apples. An example of a probability density function that follows a normal distribution:

 

     The x-axis reflects the values of the continuous random variable. The probability of a range of variables will be given by the integral of the curve for that range, or the area under the graph. Since the graph represents all possible values of your continuous random variable, the area under the entire graph is exactly equal to one. Note that the probability of a single instance occurring is infinitesimally small. For example, consider a rolling ball. With reference to a probability density function, you could determine what the probability is for the ball to stop rolling at some point between 5 seconds and 10 seconds from now. But, the probability of the ball stopping at exactly the 5th second, and not even a fraction of a millisecond earlier or later, would be infinitesimally small.  

Mathematical Statistics: Probability for Length of a Stick

A point is chosen uniformly over a 1-yard wooden stick, and a mark is made. The procedure is repeated, independently, and another mark is made. The stick is then sawn at the two marks, yielding three shorter sticks. What is the probability that at least one of those sticks is at least ½ yard long?

Solve a real linear programming question using excel solver

DO NOT round any number, and please keep at least 2 decimal places for the answers. The Weinberger Electronics Corporation manufactures four highly technical products that it supplies to aerospace firms that hold NASA contracts. Each of the products must pass through the following departments before they are shipped: wiri

Applied Probability Questions

1. Jeru is having a debate with his classmate about whether various random variables are considered discrete or continuous. Consider the shelf life of a box of "Krispy Kreme" Donuts. Jeru states that the shelf life of a box of donuts represents a discrete event because the expiration date can only occur once. Is this true or fal

Using the Mann-Whitney-Wilcoxon Test

Chapter 19 Nonparametric method The median annual income for college graduates with a bachelor's degree is $37,700 (the New York Times Almanac, 2006) Data (in thousands of dollars) for a sample of college graduates with a bachelor's degree working in the Chicago area are shown. Use the sample data to test H0: median ≤37.7 an

Dice roll experiment

What is the mean number of rolls of a die before a 1 is observed? Roll a die until a 1 is observed. Repeat this process 30 times and answer the following questions. 1. Obtain a point estimate of the mean number of rolls of a die before a 1 is observed. 2. The population standard deviation for the number of rolls before

Descriptive Statistics and Probability Distribution

In Problems 12, determine whether the distribution is a discrete probability distribution. If not, state why. #12. x P(x) 1 0 2 0 3 0 4 0 5 1 In Problems 15 and 16, determine the required value of the missing probability to make the distribution a discrete probability distribution. #16. X P(x)

Determining probability of expected birthdays

There are 25 students in a probability class. Assume that there are 365 days in each year, and that the birthrate is constant throughout the year. (a) What is the expected number of days that are the birthday of exactly one student in the class? (b) What is the expected number of days that are the birthdays of at least two s

Probability Density Function for Cost

The actual cost of a system, X, in $50,000, is predicted by the probability density function f(x) = { 0.5 - |0.5-0.25x| for 0 <= x <= 4 and 0 otherwise 1. What is the expected cost to system? 2. What is the cost Xs of system for which the probability of exceeding Xs is 0.01?

Finding the Probability Density Function

Let X be a random variable with probability density distribution given by f(x) = { x, 0 (less than or equal to) x (less than or equal to) 1, 1, 1 < x (less than or equal to) 1.5 0, otherwise. Find the probability density function of Y=10X - 4.

Probability density function: red and white chips

Urn I and urn II each have two red chips and two white chips. Two chips are drawn simultaneously from each earn. Let X_1 be the number of red chips in the first sample and X_2 the number of red chips in the second sample. Find the PDF (probability density function) of X_1 + x_2.

Examine proper probability density function.

1.Consider the following discrete probability distribution function for the variable X: X | 2 3 4 5 6 ----------------------------------------------- F(x) | .05 .4 .2 .05 .3 a. Is this a proper probability density function? How do you know? (3 pts) b. Find the mean, variance, and

Density Functions: Time takes students to finish aptitude test

1. The time it takes for a student to finish an aptitude test has the density function f(x) = c(x-1) (2-x), 1<x<2 = 0, otherwise. Here the unit of time is an hour. a) Evaluate the Normalizing constant c. b) Plot the density function c) Calculate the probability that a student will finish the test in less than 75 minut

Probability density function: Length of telemarketing phone call

The length of a particular telemarketing phone call, x, has an exponential distribution with mean equal to 1.5 minutes. a) Write down the probability density function of random variable x. b) Calculate the probability that the length of a randomly selected call will be no more than three minutes. c) Calculate the probabilit

Proportion of cars falls into what range for gas mileage

Gas mileage for both of these cars is around 17 mpg in town and 25 mpg on the highway. Suppose that the highway gas mileage rates for both of these cars are uniformly distributed over a range from 20 to 30 mpg. What proportion of these cars would fall into the 22 to 27 mpg range? Compute the proportion of cars that get more t

Marginal and Conditional Random Variables

Please see the attached file for full problem description. Let X and Y be two jointly distributed random variables having joint pdf What is the marginal pdf fX(x)? What is the conditional pdf fY(y|{X=x})? Are X and Y statistically independent? Are X and Y uncorrelated?

Probability problems based on density function

Find the probability that none of the 5 bulbs in a signal light will have to be replaced during the first 1200 hours of operation if the lifetime X of the bulb is a random variable with the density f(x)=6[0.25-(x-1.5)^2] when 1<=x<=2, and f(x)=0 other wise Where x is measured in multiples of 1000

Integration of Joint Probability Density Functions

Provide an example of how to set up the integral ranges to find a particular probability area. Let Y_1 and Y_2 have joint probability density function given by f(y_1,y_2) = 6(1-y_2) if 0 <= y_1 <= y_2 <= 1 0 elsewhere a) Find P(Y_1 <= 3/4 , Y_2 >= 1/2) b) Find P(Y_1 <= 3/4 , Y_2 <= 3/4)

Exponential probability density function

Consider the following exponential probability density function: F(x) = (1/14)e^(-x/14) for x>=0 This number represents the time between arrivals of customers at the drive-up window of a bank. a. Find f(x<=7) b. Find f(3.5<=x<=7)

Probability and Statistics Problem

The distribution function of the random variable Y is given by F(y) = {1 - 9/y2 for y > 3 0 elsewhere } Find P(X < 5) and P(Y > 8) If the joint probability density of X and Y is given by f(x, y) = {24y (1 - x - y) for x > 0, y > 0, x + y < 1 0

Normal Probability: Normal vs. Standard Normal Density Curves

Attached is a graph of a normal distribution with a mean of 2 and a standard deviation of 2. The shaded region represents the probability of obtaining a value from this distribution that is between -1 and 1. Shade the corresponding region under the standard normal density curve. (Please see the attachment).

Probability density for life span of electronic componet

Please see the attached document. Please see the attachment. Please solve the following problem: The lifetime (in hours) Y of an electronic component is a random variable with density function given by f(y) = Three of these components operate independently in a piece of equipment. The equipment fails if at least t

Probability: exponential probability density function

The lifetime (hours) of an electronic device is a random variable with the exponential probability density function: f(x)= 1/50 e ^-x/50 for x>=0 a. What is the mean lifetime of the device? b. What is the probability the device fails in the first 25 hours of operation? c. What is the probability the device operates 100

Rayleigh density

1. A density function sometimes used by engineers to model length of life of electronic components is the Rayleigh density, given by if 0 < y1 <infinity, 0 otherwise. Assume Y1, Y2, ...Yn is a random sample from a Rayleigh distribution. a. If Y has the Rayleigh density, find the probability density for U = Y2. b. Use part a

Mathematical Statistics

1. (Adapted from Larson, Intro. to Probability Theory, 1969) A nursery specializes in the installation of circular flower beds. When laying out the circle, a workman puts a peg in the center and cuts a length of rope (already tied in a loose loop to the stake) equal to the radius of the desired circle, and uses this to mark ou

Continuous Random Variable and Probability Density Function

Let X denote a continuous random variable with probability density function f(x) = kx^3/15 for 1≤ X ≤ 2. a. Determine the value of the constant k. b. Determine the probability that X > 1.5. c. Determine the cumulative distribution function F(x) and state the values of F(x) at x = 0.5, 1.5, and 2.5.