5. Distributions (a) Write down the formula for the density of the exponential distribution. (b) What does it mean that a random variable coming from an exponential has no memory? (c) Prove that an exponentially distributed random variable has no memory.

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Solve by MATLAB

Consider a gas in a piston-cylinder device in which the temperature is held constant. As the volume of the device was changed, the pressure was mecas- ured. The volume and pressure values are reported in the following table: Volume, m Pressure, kPa, when I= 300 K 2494 1247 831 4 623 5 499 416 (a) Usc lincar interpolation to estimate the pressure when the volume is 3.8 m. (b) Usc cubic splinc interpolation to cstimate the pressure when the vol- ume is 3.8 m. (c) Usc lincar interpolation to cstimate the volume if the pressure is meas- ured to be 1000 kPa. (d) Usc cubic splinc interpolation to cstimate the volume if the pressure is mcasured to be 1000 kPa. 4.

Solve in R programming language: 5) Use R to calculate and simulate with the exponential distribution as follows. (a) For an exponential random variable X with λ = 4, simulate 1000 independent exponential random variables by using the R function rexp(n, λ). Calculate the mean and variance of this sample. (b) Compare the empirical results from part (a) with the distribution mean 1/λ and the distribution standard deviation 1/λ.

Please help step to step with Program R (CS) with explanation and final code for understanding thank you.

Please help step by step with an explanation for Program R (CS) with a final code for understanding thank you.

Solve both the questions!!!

The flight of a model rocket can be modeled as follows. During the first 0.15 s the rocket is propelled upward by the rocket engine with a force of 16 N. The rocket then flies up while slowing down under the force of gravity. After it reaches the apex, the rocket starts to fall back down. When its downward velocity reaches 20 m/s, a parachute opens (assumed to open instantly), and the rocket continues to drop at a constant speed of 20 m/s until it hits the ground. Write a program that calculates and plots the speed and altitude of the rocket as a function of time during the flight. SOLVE WITH MATLAB PLEASE

Please help step to step with Program R (CS) with explanation and final code for understanding thank you.