Problem2 2.1) Write down the mathematical definition of each of the following. (

Question

Problem2 2.1) Write down the mathematical definition of each of the following. (Be precise in your definitions. Each definition should be a mathematical statement.) (a) A random variable (b) The pdf of a random variable. (Specify whether the random variable is discrete or continuous.) (c) The pmf of a random variable. (Specify whether the random variable is discrete or continuous.) (d) The cdf of a random variable. (Specify whether there are any restrictions on whether the random variable is discrete or continuous.) (e) the expected value of a random variable. (Give separate expressions for discrete and continuous random variables.) () the expected value of a function of a random variable. (Give separate expressions for discrete and continuous random variables.) 2.2) Write down the mathematical definition of each of the following random variables. (Each definition should be a mathematical statement.) For any parameters of the distribution, explain the meaning of each parameter. Specify whether each of these is a discrete or continuous random variable (a) a Bernoulli random variable (b) a Binomial random variable (c) a geometric random variable (d) a Poisson random variable (e) a uniform random variable () an exponential random variable (g) a normal (or Gaussian) random variable

Explanation / Answer

a) Random variable is a function that assigns values to each of an experiment's outcomes.

b) If X is a continuous variable and fX(x) is a continuous function of X, fX(x) dx gives the probability of the event that X lies in the interval

(x-1/2dx) and (x+1/2dx), f(x) is called probability density function.

c) A probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value that is pi = P(X = xi) =P(xi)

such that p(xi) > 0 for all i.

d) A function FX(x) of a random variable X for a real value x giving the probability of the event (X x) is called cumulative distribution function. Symbolically FX(x) = P(X x).

Obviously X lies in the interval (- ,x)

e) The expected value of a random variable is simply the long run average of this variable over an indefinite number of samples.

If X is a discrete random variable that takes on the values x1 ,x2,…xn with probabilities p1,p2,…p3respectively, the expected value of X is given by:

E(X) = ni=1 pi xi

If X is a continuous random variable with p.d.f f(x) the expected value of X is

E(X) = x. f(x) dx; axb

f) If X is a discrete random variable that takes on the values x1 ,x2,…xn with probabilities p1,p2,…p3respectively, the expected value of a function H(x) is given by:

E{H(x)} = H(x) p(x)

If g(x), a function of X is a continuous random variable and E{g(x)} exists,then

E[g(x)] = g(x) f(x) dx

2.2)

a) A random variable X, marked by onlt two values 1 and 0 with probability of occurance p and q(=1-p) respectively where p(x=1)=p and P(x=0)=q, is called Bernoulli variate and its distribution is called Bernoulli distribution. The probability distribution of x is,

f(x) = px q1-x where 0 p 1 and x =0 or 1

The only parameter is p.

b)   Let n(finite) Bernouli trails be conducted with probability p of success and q of a failure. The probability of x successes out of n Bernoulli trails is given by:

f(x) = nCx px qn-x ; x=0,1,2…n , 0 p 1, p+q=1

The parameters are n and p.

c) if there are a number of trails such that the probability of success p at each trail remains the same, the probability that there are x failures before the first success is given by p.qx. Hence, a random variable X which can take only positive integer values is said to follow geometric distribution if its probability mass function is

f(x) = P(X=x) = p.qx. for x =0,1,2… , 0 p 1

The only parameter is p.

d) If dichotomous variable X is such that the constant probability p of success for each trail is very small and the number of trails n is indefinitely large and np= is finite, the probability of x successes is given by the probability mass function,

P(X=x) = e- x / x!

The only parameter is .

e) A random variable is said to follow continuous uniform distribution in an interval (a,b) if its density function is constant over the entire range of the variable X. its functional form is, f(x) = 1/(b-a) if axb

             0         otherwise

f) A continuous random variable X is said to follow exponential distribution if for any positive value µ,it has the probability density function,

f(x) = µ e-µx for µ >0 ,x>0

The only parameter is µ.

g) A continuous random variable X is said to follow X is follow a normal distribution with mean µ and variance 2, if its probability density function is

f(x) = 1/(2) exp( -(x-µ)2/(22)

The two parameters are µ and 2.

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