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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


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.


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)


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


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?


Subject: Probability Details: You appear on a game show and for your prize the host lets you choose one of three doors. Behind one door is a new car; behind each of the other doors is a goat. You choose a door. The host, who knows what's behind each door, then opens another door, which has a goat. He then asks if you want to p

Exponential random variable

I)Let X be an exponential random variable with mean 1,and Y = exp^(X/2): a)find F(y) b)Evaluate E(Y) c)Evaluate E[(Y^2)/(1+(X^2))] II)The random variable X is uniformly distributed on the interval [1,3].Find the probability density function fy(y)of the random variable Y=2X+5

Simple Random Survey, Proportion, Probability, Mean, Standard Deviation

According to hypothetical information, 18% of all Americans have hazel colored eyes. A SRS of 25,000 Americans are surveyed and asked about their eye color (assume each of the Americans in the sample has an 18% chance of having hazel colored eyes). a) Determine the mean and standard deviation of this sample of Americans that


The scores of students taking the Scholastic Aptitude Test (SAT) are normally distributed with a mean u = 1035 and a standard deviation o = 131. a) Determine the probability that a simple random sample of 15 students would earn an average score of 1000 or less. b) Determine the probability that a simple random sample


Suppose you pay $10 to play a card game with the following rules: You draw 2 cards out of a standard deck of 52 cards without replacement. If the 2 cards are of the same suit (i.e. both clubs, both diamonds, etc) you win $15 (a $5 profit). If the 2 cards are the same value (i.e. both jacks, both 2's, etc) you win $18 (a

Statistics - molecule speed

This is for question #3 on the attached page... You will need the following fact for Question #3. integral_{-infinity}^{+infinity} e^(-x^2/2 sigma^2) dx / sqrt(2*pi*sigma^2) = 1 This is in Section 5.4: it just says that the function e^(-x^2/2 sigma^2) / sqrt(2*pi*sigma^2) is a density (it is the "famous" normal de

Statistical probability

The speed of a molecule in a uniform gas at equilibrium is a random variable whose probability density function is given by f(x) = (ax2e&#8722;bx2 x>_0 { 0 x < 0 where b = m/2kT and k, T and m denote, respectively, Boltzmann's constant, the absolute temperature, and the mass of the molecule. Evaluate a i

Cumulative distribution function (same as 26793)

A cdf Fx is stochastically greater than a cdf Fy if (1) Fx(t)<= Fy(t) for all t, and (2) there exists some t for which Fx(t) < Fy(t) (a)show that if Fx is the cdf of X and Fy is the cdf of Y, then (1) P(X>t) >= P(Y>t) for all t and (2) P(X>t) > P(Y>t) for some t.(in other words, X tend to be bigger than Y) Give an example.

Mean and variance

How to calculate mean and variance of geometric distribution (by differentiating moment-generating function)?


From 5 statisticians and 6 economists, a committee consisting of 3 statisticians and 2 economists is to be formed. How many different committees can be formed if: (a) there are no restrictions (b) 2 particular statisticians must be on the committee (c) 1 particular economist cannot be on the committee Answers: (a) 150 (

Poisson distribution

2. Mistakes made by data entry clerks follows a Poisson distribution with a mean 0.8 per hour. You just gave someone applying for your data entry vacancy an hours worth of data entry material. Answer the following questions. A) What is the probability they will not make any mistakes? B) What is the probability they will