Aparticle’s velocity (in m/s) in a gas is a random variable (let it be v ) with pdf fv (v) = av2 e −bv, v > 0, where b is a constant that depends on the temperature of the gas and the mass of the particle. answer the following questions. (a) what should a be for this to be a valid pdf as a function of b? (b) what is the probability the velocity of the particle is between 10 and 100 meters per second? (c) the kinetic energy of a particle is w = 1 2mv 2 , where v is the speed. what is the pdf of w? (d) what is the mean kinetic energy of a particle as a function of constant b?

Neptune is the eighth planet from the sun. it was the first planet to get its existence predicted by  mathematical calculations  before it was actually seen through a telescope on sept. 23, 1846. irregularities in the orbit of  uranus  led french astronomer alexis bouvard to suggest that the gravitational pull from another celestial body might be responsible. german astronomer johann galle then relied on subsequent calculations to   spot neptune via telescope. previously, astronomer galileo galilei sketched the planet, but he mistook it for a star due to its slow motion. in accordance with all the other planets seen in the sky, this new world was given a name from greek and roman mythology — neptune, the roman god of the sea.