Statistics: Queueing Problem

A fast food outlet has an average of 8 cars at the drivethrough during "lunch rush" 11am-1pm. On average, 2 cars per min. arrive at the resaurant parking lot, and consider the drivethrough but 25% of the time, an arriving car does not actually enter the drive-through line (i.e. it "balks"). Assume no car enters the line without ...continues

Probability: Moment Generating Functions and Poisson Process

1.) Let X be a discrete random variable with probability mass function Pr {X=k} = c(1+ k^2) for k= -2, -1, 0, 1, 2. a) Determine c. b) Determine Pr {X <= 0} c) Determine the mean of X d) Why is the previous answer fairly obvious? e) Determine the variance of X f) Compute Pr {X=2 | X >= 0} g) Determine the moment genera ...continues

Random Variables : Continuous R.V., Exponenetial, R.V, Mean and Variance

3) Let X be a continuous random variable with probability density function f(s)= c(1 + s^2) for -2 <= s <= 2. a) Determine c b) Determine Pr {X <= 0} c) Determine the mean of X d) Why is the previous answer fairly obvious? e) Determine the variance of X f) Compute Pr {X = 2 | X = 0} g) Determine the cumulative distribut ...continues

density function, variance

6) Suppose we have an aisle with storage racks on both sides of the aisle. The aisle is 100 feet long. A worker is stationed at one end of the aisle. The worker needs to retrieve an item from storage. Assume that all storage locations are equally likely. Let L be the distance that the worker needs to walk along the aisle to reac ...continues

Functions : Proof by Induction

Let n be a natural number, and let f(x) = x^n for all x are members of R. 1) if n is even, then f is strictly increasing, hence one-to-one, on [0,infinity) and f([0,infinity)) = [0,infinity). 2) if n is odd, then f is strictly increasing, hence one-to-one, on R and f(R) = R. Prove that f is strictly increasing by indu ...continues

Real Analysis Problem

The inverse cosine function has domain [-1,1]and range [0, pi]. Prove that (cos^-1)'(x) = -1/ sqrt(1-x^2). This needs to be proved from a real analysis point of view not a calculus.

Real Analysis: Proof by induction that a polynomial of degree n>0 has at most n roots.

A polynomial of degree n>0 has at most n roots. (A root of a function is a point at which the function has value 0.) I need a proof by induction to show this.

MacLaurin Series And Laplace Transforms : Absolute Convergence

Find MacLaurin Series for the given function f. Use the linearity of the Laplace Transform to obtain a series representation L(f)=F(s) Determine 5 values for which the series converges absolutley (and uniformly). Also show the Laplace transform exists, i.e. that it has exponential order alpha. Here are the functions. A) f ...continues

Combinations : Problem Solving With Dice

Please see the attached file for the fully formatted problems. Question1: How many dots at the outer sides of the dice. Given view is from the top of these four dice? Question2: In the newly formed shaped below do you think answer is the same or different from above? What does your intuition say without investigating? Q ...continues

Real Anaylsis - Mean Value Theorem

Define f : [ -2,2]---> R by f(x) = x^3 - 3x + 3. Find all numbers p in [-2,2] that satisfy the conclusion of the Mean Value Theorem. How do we know we have them all?