Question:

X is a random variable with an exponential distribution with rate ? = 0.7 Thus the pdf of X is f(x) = ? e??x

for 0 ? x where ? = 0.7.

a) Using the f(x) above and the R integrate function calculate the expected value of X.

b) Using the f(x) above and the R integrate function calculate the expected value of X0.2

.

c) Using the dexp function and the R integrate command calculate the expected value of X.

d) Using the pexp function find the probability that .33 ? X ? 1

e) Calculate the probability that X > 0.15 by using the pexp function

f) Calculate the probability that X is at least 1.5 more than its expected value. Use the pexp function

g) Copy your R script for the above into the text box here.

Spring ECE 103 Engineering Programming Requirements Write a C program that meets the following specifications: . The program prompts the user to enter the following information in this order. The curvature Ri (in cm) of mirror 1 (located at plane position z, ). o The curvature Rz (in cm) of mirror 2 (located at plane position z, ). The distance L (in cm) between the centers of the two mirrors. The wavelength A (in nm) of light in the lasing medium. . The program calculates and displays the following values: Minimum spot size w. (in cm). Position Z, (in cm) of mirror 1. . Position Z, (in cm) of mirror 2. o Beam angular spread O (in radians). Suppose the laser is pointed at a satellite in a geostationary orbit a distance 35786 km away. Calculate and display an estimate of the beam's width (in km) at the satellite's location. Be aware that you may need to perform some unit conversions. Define a macro for the value of a. For consistency, use 3.141592654. Display your calculated results to three (3) decimal places. Hint: To match the sample output, consider using the &g option. Your program must conform to either the ISO C99 or ISO C90 standard. Save your program using this filename: hw2. cThe Dirac delta function in three dimensions can be taken as the improper limit as a -0 of the Gaussian function 1 D(a; x, y, z) = (21) 3/20 3 exp 202 ( 12 + 12 + 2 2 ) Consider a general orthogonal coordinate system specified by the surfaces u = constant, v = constant, w = constant, with length elements du/U, do/V, dw/W in the three perpendicular directions. Show that 6(x - x') = 8(u - u') 8(v - v') 8(w - w') . UVW by considering the limit of the Gaussian above. Note that as a - 0 only the infin- itesimal length element need be used for the distance between the points in the exponent.5. {8 pts} 011 one street, a drivers speed just before an accident is unifome distributed on [15, 25]. Given the speed, the resulting loss from the accident is exponentially distributed with mean of 1.25 times the speed. Calculate the variance of a loss due to an accident on that street