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Find the cumulative distribution function

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1. Suppose at each step, a particle moving on sites labelled by integer has three choices: move one site to the right with probability p (where 0 ≤ p ≤  1/2 ), move one site to the left with probability 1 - 2p or stay where it is with probability p.

370_market.png

 

The particle starts at position 0 and moves two steps.

(a) Let f(p) be the probability that the particle is at position 0 after moving two steps. Find f(p) in terms of p and evaluate the expression (correct to 2 decimal places) when p = 0:3.

(b) Let g(p) be the probability that the particle is one site away from position 0 after making two steps (positions +1 and -1 both qualify as being one site away from position 0). Find g(p) in terms of p and evaluate the expression (correct to 2 decimal places) when p = 0:3.

(c) Let h(p) be the probability that the particle is two sites away from position 0 after making two steps. Find h(p) in terms of p and evaluate the expression (correct to 2 decimal places) when p = 0:3. 

(d) Find the value of p that minimizes f(p). 

(e) Find the value of p that maximizes g(p).

2. The continuous random variable X has cdf (cumulative distribution function) given by

401_market2.png

(a) Find the value of k. 

(b) Find E(X). 

(c) Find the median of X. 

(d) Find the standard deviation of X, correct to 2 decimal places.

(e) Denoting E(X) by μ and the standard deviation of X by , nd P(X < (σ)).

3. The velocity V of a moving object has a probability density function (pdf) shown in the diagram:

462_market1.png

(a) Find the cumulative distribution function (cdf) of V . 

(b) In Physics, we know that the relationship between kinetic energy K and the ve- locity V of a moving object with mass m (m is xed in this question) is:

K = 1/2mV2

If V has pdf as given in the diagram above, nd the pdf of K. 

 

 

 

 

 

 

 

 

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