Problem 1 For Gaussian distribution ρ (x)-ae Find: (1) Constant a; (2) <x> , <x> and...
*Problem 1.6 Consider the Gaussian distribution A(x-a)2 where A, a, and λ are constants. (Look up any integrals you need.) (a) Use Equation 1.16 to determine A. b) Find (x),(x-), and σ (c) Sketch the graph of ρ (x).
Problem 2 Consider the wave function Where a, λ ω are positive constants. (a) Normalize (b) Determine the expectation values ofx and x; (c) Find the standard deviation ofx. Sketch the graph of 1992, as a function ofx, and mark the points (<x> + σ) and 〈X>-07, to illustrate the sense in which σ represents the "spread" in x, what is the probability that the particle would be found outside this range?
its applied engineering data analysis course Q4 X is the diameter (in mm) of tires, normally distributed with mean 575 and a standard deviation of 5 SKETCH THE AREA of P(575 < X < 579) in both X and Z and find P Find the diameter x such that there are only 1% tires longer than this diameter ie. P[X>x] 0.01 Find the (diameters of) tires that have most extreme 5% diameters. a. b. C.
4. Let X be a continuous random variable with probability density function: x<1 0, if if| if x>4 f(x) = (x2 + 1), 4 x 24 0 Find the standard deviation of random variable X.
x-μ 6. Hint: use the formula*. If X falls within a range, transform both lower and upper limits. Let X be normally distributed with mean deviation σ 4 a. Find P(X30). b. Find P(X> 2), c. Find P(4 <X< 10). d. Find P(6 <X< 14) 10 and standard
xercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < 1 . Check that if X Z and Y-: ρΖ+ VI-P" W then the pair (X, Y) has standard bivariate normal distribution with parameter p. Hint. You can use Fact 6.41 or arrange the calculation so that a change of variable in the inner integral of a double integral leads to the right density function.
7. Find the solution of the heat conduction problem 100uzz = ut, 0 < x < 1, t > 0; u(0,t) 0, u1,t 0, t>0; In Problem 10, consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0°C for all t0. Find an expression for the temperature u(,t) if the initial temperature distribution in the rod is the given function. Suppose that a
Suppose a joint probability density function for two variables X and Y is given as follows: {24x0, if 0 < x < 1,0 < y < 1 f(x, y) = otherwise Please find the probability p (w > 1) =? 3
(I point) f(z)-,2+1-1 < z < 0 (i) find P(-0.5sX<0.25). (a) Find the cumulative distribution function F(z). Fill in the blanks below. F(z) EE when x when when when x> (b) Evaluate P(Xc0.75X20.25) (c) 35% of the time, X exceeds what value? (d) l Estimate the location of the mean/expected value of X. Once you have done so, find the E(X)
Problem 2 If the cumulative distribution function of X is given by o F(b) = b<0 0<b<1 1<b<2 2<b<3 3<b<3.5 b> 3.5 1 calculate the probability mass function of X.