Question

(a) (i) Define a probability measure. (ii) Consider an experiment consisting of choosing a point at random from the interval [0, 1]. Let A be a subset of [0, 1], show that P(A) is a probability measure on [0, 1]. (b) The random variable X has probability density function given by f_X(x) = c(1 - x^2), -1 ? x ? 1 0, otherwise where c is a constant. (i) Show that c = 3/4. (ii) Find the distribution function of X. (c) The lifetime of a special type of battery is a random variable with mean 40 hours and standard deviation 20 hours. A battery is used until it fails, at which point it is replaced by a new one. Assuming a stockpile of 25 such batteries the lifetime of which are independent, find the probability that over 1100 hours of use can be obtained. (d) If X and Y are independent random variables both uniformly distributed on (0, 1). Find the probability density of Z = X + Y. [3, 5, 3, 4] (a) (i) State and prove Chebyshev's inequality. (ii) Let the random variable X have a binomial (5, 0.34) distribution. Use Chebyshev's inequality to find the bound on P(|X - ?| < 3). (b) If X and Y are independent Poisson random variables with respective means ?1 and ?2, calculate the conditional expected value of X given that X + Y = n. (c) The lifetime of a light-bulb produced by Savenda Illumination Company is known to have an exponential distribution with parameter ?. However, the company has been experiencing quality control problems and on any given day, the parameter ? is a random variable uniformly distributed in the interval [0, 0.5]. Find the probability that the lifetime of a randomly chosen light-bulb is exactly n. (d) Find the moment generating function of a Poisson random variable X with parameter ?. Hence, find the mean and variance of X. [4, 4, 3, 4] (a) If X and Y are independent gamma random variables with parameters (?, ?) and (?, ?) respectively. Compute the joint density of U = X + Y and V = X/(X+Y). (b) Let X be a random variable. Prove that the set function P_X(B) = P[X^-1(B)] is a probability measure on R. (c) (i) State and prove Markov's Inequality. (ii) Suppose that a non negative random variable X has mean 6, use Markov's Inequality to find a bound on P(X ? 20).

          (a) (i) Define a probability measure.
(ii) Consider an experiment consisting of choosing a point at random from the interval [0, 1]. Let A be a subset of [0, 1], show that P(A) is a probability measure on [0, 1].
(b) The random variable X has probability density function given by
f_X(x) = 
c(1 - x^2), -1 ? x ? 1
0, otherwise
where c is a constant.
(i) Show that c = 3/4.
(ii) Find the distribution function of X.
(c) The lifetime of a special type of battery is a random variable with mean 40 hours and standard deviation 20 hours. A battery is used until it fails, at which point it is replaced by a new one. Assuming a stockpile of 25 such batteries the lifetime of which are independent, find the probability that over 1100 hours of use can be obtained.
(d) If X and Y are independent random variables both uniformly distributed on (0, 1). Find the probability density of Z = X + Y.
[3, 5, 3, 4]
(a) (i) State and prove Chebyshev's inequality.
(ii) Let the random variable X have a binomial (5, 0.34) distribution. Use Chebyshev's inequality to find the bound on P(|X - ?| < 3).
(b) If X and Y are independent Poisson random variables with respective means ?1 and ?2, calculate the conditional expected value of X given that X + Y = n.
(c) The lifetime of a light-bulb produced by Savenda Illumination Company is known to have an exponential distribution with parameter ?. However, the company has been experiencing quality control problems and on any given day, the parameter ? is a random variable uniformly distributed in the interval [0, 0.5]. Find the probability that the lifetime of a randomly chosen light-bulb is exactly n.
(d) Find the moment generating function of a Poisson random variable X with parameter ?. Hence, find the mean and variance of X.
[4, 4, 3, 4]
(a) If X and Y are independent gamma random variables with parameters (?, ?) and (?, ?) respectively. Compute the joint density of U = X + Y and V = X/(X+Y).
(b) Let X be a random variable. Prove that the set function P_X(B) = P[X^-1(B)] is a probability measure on R.
(c) (i) State and prove Markov's Inequality.
(ii) Suppose that a non negative random variable X has mean 6, use Markov's Inequality to find a bound on P(X ? 20).

(a) (i) Define a probability measure.
(ii) Consider an experiment consisting of choosing a point at random from the interval [0, 1]. Let A be a subset of [0, 1], show that P(A) is a probability measure on [0, 1].
(b) The random variable X has probability density function given by
fX(x) =
c(1 - x^2), -1 ? x ? 1
0, otherwise
where c is a constant.
(i) Show that c = 3/4.
(ii) Find the distribution function of X.
(c) The lifetime of a special type of battery is a random variable with mean 40 hours and standard deviation 20 hours. A battery is used until it fails, at which point it is replaced by a new one. Assuming a stockpile of 25 such batteries the lifetime of which are independent, find the probability that over 1100 hours of use can be obtained.
(d) If X and Y are independent random variables both uniformly distributed on (0, 1). Find the probability density of Z = X + Y.
[3, 5, 3, 4]
(a) (i) State and prove Chebyshev's inequality.
(ii) Let the random variable X have a binomial (5, 0.34) distribution. Use Chebyshev's inequality to find the bound on P(|X - ?| < 3).
(b) If X and Y are independent Poisson random variables with respective means ?1 and ?2, calculate the conditional expected value of X given that X + Y = n.
(c) The lifetime of a light-bulb produced by Savenda Illumination Company is known to have an exponential distribution with parameter ?. However, the company has been experiencing quality control problems and on any given day, the parameter ? is a random variable uniformly distributed in the interval [0, 0.5]. Find the probability that the lifetime of a randomly chosen light-bulb is exactly n.
(d) Find the moment generating function of a Poisson random variable X with parameter ?. Hence, find the mean and variance of X.
[4, 4, 3, 4]
(a) If X and Y are independent gamma random variables with parameters (?, ?) and (?, ?) respectively. Compute the joint density of U = X + Y and V = X/(X+Y).
(b) Let X be a random variable. Prove that the set function PX(B) = P[X^-1(B)] is a probability measure on R.
(c) (i) State and prove Markov's Inequality.
(ii) Suppose that a non negative random variable X has mean 6, use Markov's Inequality to find a bound on P(X ? 20).

Added by Katanekwa S.

Question

Please give Ace some feedback