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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.  

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)

Density Functions: Time it takes students to finish an 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

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


The number of failures of a unit from contamination particles on the unit is a Poisson random variable with a mean of 0.02 failures per hour. a.) What is the probability that the instrument does not fail in an 8-hour shift? b.) What is the probability of at least one failure in a 24-hour day? Samples are defective in 1% o

Normal probability calculation

#1 The average number of customer complaints at the Full Moon Motel is five per day. Find the probability that on a typical day, the motel will receive a) eight complaints b) at least two complaints #2 Demand for a product is normal, with mean being 400 and standard deviation being 25 units. Find a) Pr (demand >

Probability, Cumulative Distribution Function

Let X be a random variable with probability density f(x) = c(1-x^2), if -1<x<1 = 0, if otherwise Determine the value of the constant , and find the cumulative distribution function of X

Probability and Marginal Density Function

Let X and Y have joint probability density function f(x,y) (s,t) = 20e^-(4s+5t) for 0 < s, and 0 < t. Find (a) Pr{X = Y} (b) Pr{min(X,y) > 1/5}, (c) Pr {X < Y}, (d) the marginal probability density function of X, and (e) E[XY].

Demand, Revenue and Supply Functions

1. What is true for the function: q=D (p) =-.53p+1283, where p=95. a) Demand is instantaneous b) Demand is marginal c) Demand is inelastic d) Demand is elastic e) Demand is unit elastic 2. In a sample of 50 individuals, 7 said they would pay up to $100 for World Series Tickets. Of the people surveyed, what is

Joint and Marginal Probability Density Functions of Independent Variables

Let X, Y be independent, standard normal random variables, and let U = X + Y and V = X - Y. (a) Find the joint probability density function of (U, V) and specify its domain. (b) Find the marginal probability density function of U and V specifying the domain in each case. (c) Explain why U and V are independent Joint probab

The problem is from probability class.

#4. The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by f (x) = 10/x2 x > 10 f (x) = 0 x &#8804; 10 (a) Find P{X > 20}. (b) What is cumulative distribution function of X? .. (see attachment)

Joint Density

3. The joint probability density function of X and Y is given by f(x, y) =6/7 (x2 + xy/2 ), 0 < x < 1, 0 < y < 2. (a) Verify that this is indeed a joint density function. (b) Compute the density function of X. (c) Find P(X > Y ). (d) Find P(Y > 1/2|X < 1/2). (e) Find E(X). (f) Find E(Y ). Please see attachment for

Density Function Question

The density function of a random variable X is f(x) = e^-x x >= 0 = 0 otherwise Find (a) E(X), (b) E(X²), (c) E{(X-1)²}. Please see attached for full question.

Probability Density Function - Complex Gaussian Noise

Referencing the attached: NOTE: The solution for part B is highlighted within the attached file. Re-stated here it is: p(I) = ( 1 / < I > ) exp ( - I / < I > ) I'm not sure how. Part a, is essentially: Integral (infinity, 0) p(I, theta) di. which according to my calculation equals: - 1 --