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Probability

Uniform and Exponential Distributions

A) A fire station is to be located along a road of length A, A<&#8734;. If fires will occur at points uniformly chosen of (0, A), where should the station be located so as to minimize the expected distance from the fire? That is, choose a so as to minimize E[|X-a|] when X is uniformly distributed over (0,A). b) N

Probability Questions

A professor grades homework by randomly choosing 5 out of 12 homework problems to grade. a) How many different groups of 5 are there from the 12 problems? b) Jerry did only 5 problems of one assignment. What is the probability that the problems he did comprised the group that was selected to be graded? c) Silvia did 7 pro

Probability Statistic Problems

Sample Problems - Please show all your work. QUESTION 1 Suppose that you are working for an energy company that distributes natural gas to residents in a city. As the chief economic analyst, you are asked to analyze the demand for natural gas in your city. Let X denote the random monthly demand for natural gas in millions of

Probability

1) Records show that 30% of all patients admitted to a medical clinic fail to pay their bills and that eventually the bills are forgiven. Suppose n = 4 new patients represent a random selection from the large set of prospective patients served by the clinic. Find these probabilities: a) All the patients' bills will eventually h

Basic Probability Problems

1) A shipping company knows that the cost of delivering a small package within 24 hours is $14.80. The company charges $15.50 for shipment, but guarantees to refund the charge if delivery is not made within 24 hours. If the company fails to deliver only 2% of its packages within the 24 hour period, what is the expected gain per

Normal Probability Distribution.

The goal at U.S. airports handling international flights is to clear these flights within 45 minutes. Let's interpret this to mean that 95 percent of the flights are cleared in 45 minutes, so 5 percent of the flights take longer to clear. Let's also assume that the distribution is approximately normal. a) If the standard dev

Marginal Distribution for Random Variable

Let X1 and X2 be two independent random variables having Poisson Dist. with parameters mew(subscript1)=2 and mew(subscript 2)=3 respectively. Then the marginal distribution for the random variable X2 is : a.) h(X2) = e^-2*2^Xz/X2!, x2=0,1,2,3,... b.) h(X2) = e^-5*5^Xz/X2!, x2=0,1,2,3,... c.) h(X2) = e^-3*3^Xz/X2!, x2

Cumulative Distribution Function #2

A lifespan in hours for an electrical component is a random variable x, with cumulative dist. funct.: F(x) = {1-e^-x/100 for x>0 0 elsewhere Determine probability function x for x: Which of the Following Answers: a.) F(x) = {100e^-x/100 for x>0 0 elsewhere b.) F(x) = {1-1/100

Probability Problems

I.) The number of construction projects on a college campus follows Poissan's dist. with a mean=3. The probability that exactly two projects are currently taking place is: 1.) 0.4230 2.) 0.224 3.) 0.00 4.) 0.1990 J.) In an assembly line with robots, a particular component can be installed in 90 seconds if holes a

Probability Problems

F.) A rock crushing company has 3 plants, all receiving blasted rock in bulk. The amount of rock that can be crushed by one of the plants in one day can be modeled by an exponential distribution. The mean amount of rocks that can be crushed per day by each plant is 4 tons. Assume plants operate independently of one another.

Moment generating function and Poissan's Dist.

D.) Suppose X1 and X2 are independent exp. random variables, each with mean data and y=X1 and X2. What is the moment generating function for the random variable y? Which choice below is right: 1.) My = (1-t/beta)^-2 2.) My = (1-t/beta)^2 3.) My = (1-beta*t)^-2 4.) My = (1-beta*t)^2 E.) Let X1 and X2 be two di

Defective Parts w/o replacement and Group size

Two separate questions below: A.) A bin contains 20 fuses of which 5 will be defective. If 2 fuses are selected at random without replacement, what is the probability that at most, one is defective?(Please show equation) B.) From a group of 5 men and 6 women, how many committees of size 3 are possible with 2 men and one w

Order Statistics

Let X1, X2, X3 be random sample from a distribution of the continuous type having pdf f(x)=2x, 0 <x < infinity, 0 elsewhere. Compute the probability that the smallest of X1, X2, X3 exceeds the median of the distribution.

Statistics: A Coin Toss

A fair coin is tossed 3 times. Let X be the number of heads on the third tossing and let Y be the total number of heads. Questions: a) Find the joint pdf of X and Y. b) Find the marginal pdf of X. Compute E(x) and V(x) - does X follow classical distribution? c) Find the marginal cdf of Y. Compute E(y) and V(y) - does Y

Density Function : Median and Quartiles

The density function of a continuous variable x is given by f(x) =x; 0&#8804;x<1 and f(x) = 2-x; 1&#8804;x<2 a) Find p(1/2&#8804;x&#8804;3/2) b) Find the median and quartiles of this distribution

Pdf or cdf

A manufacturer of computer printers claims that only 0.55% of all printers they produce fail out of the factory. As a computer store buyer, you purchase 150 of this company's printers. What is the chance that you get at least one defective printer out of this purchase? P(X is greater than or equal to 1)=____________ pleas

Probability

A jar contains marbles of different colors: 4 white, 6 black, 6 red, and 4 blue. On two random drawings, without replacement, find the probability that the first is black and the second is white. Assume that the draws are independent.

Probability

Take a deck of cards. A deeck has 52 cards with two major colors(Red and Black), with 26 cards each. Red colors comes in two shapes(heart and diamond), while black color shapes are called clubs and spades. Find the probability of obtaining Red or Queen. That is, find Probability(Red or Queen).

Probability calculations for rolling of two dice

Two dice are rolled. Observe their output space and do the following: a) Probability ( sum of numbers making at most 8) b) Probability ( sum of numbers exceeds 8) c) Probability ( both dice having equal odd numbers) d) Probability ( even numbers on both)

Probability

The animal colony in the reseach department contains 20 male rats and 30 female rats. Of the 20 males, 15 are white and 5 spotted. Of the 30 females, 15 are white, and 15 are spotted. Suppose that you randomly select 1 rate from this colony: a)Find the Joint probability table including the marginal probability. b) what is th

Probability of a Container Containing a Certain Weight

The weight of the food packed in certain containers is a normally distributed random variable with a mean weight of 400 pounds and the standard deviation of 4 pounds. Suppose that the container is picked at random. Find the probability that it contains: a) more than 410 pounds b) less than 398 pounds c) between 391 and 398

Probability problem...

How many ways can 6 distinguishable balls be placed in 5 boxes such that there are 2 balls in the first box, and one in all remaining boxes?

Probability bounds using Chebychev's inequality

Many people believe that the daily charge of a price of a company's stock on the stock market is a random variable with mean 0 and variance {see attachment}. That is, if Yn represents the price of the stock on the n-th day, then (see equation in attachment( where X1,X2,..., are independent and identically distributed random var

Normal Probability Distributions

Suppose the lifetime income of all high school grads is normally distributed and that the lifetime income of all college grads is normally distributed. Suppose that the difference in the means is known to be $600,000 and that the standard deviation of the high school graduates is $300,000 and the standard deviation of the colle

Probability

The Skateworld Company operates ice rinks in several major cities throughout the United States. During each session of open skating, one customer is selected at random to receive a free pass for a future open skating session. At a recent session there were 150 males and 130 females skating. What is the probability that the pe

Random Variables

Let X1,X2... be a sequence of independent and identically distributed continuous random variables. Define the random variable ... (a) Compute the p.m.f of N by first computing P(N [less than or equal to] n) (b) Show that E(N)=e *(Please see attachement for complete problem)

Random number sets

1. Choose a number at random from the set of numbers from the set of numbers {1,2,3,4,5}. Now choose a number from the subset {1,...,X). Call this second number Y. a) Find the joint p.m.f. of X and Y b) Find the conditional mass function of X given that Y = i. Do it for i = 1,2,3,4,5 c) Are X and Y independent? Why?