####### R Code for Sketch of PDF of X
########
x = seq(1:20)
f = exp(-(x-1))
plot(x, f, lwd =2, type = "l", col = "brown", main = " PDF of
X")
A probability distribution function for a random variable X has the form Fx(x) = A{1 -...
2-2.3 A probability distribution function for a random variable has the form F,(x) = A(1-exp[-(x-1)) 1 < x < oo -00<xs1 a) For what value of A is this a valid probability distribution function? b) What is Fx (2)? c) What is the probability that the random variable lies in the interval 2 X < 00? d) What is the probability that the random variable lies in the interval 1 <X s3?
[Total Marks: 301 ={} Question 1 A random variable X has a probability density function as defined below. (x + 1 -1<x<0 fx(x) = (-x+1 0<x< 1 Find the following: a) The cumulative distribution function of X, Fx(x). b) P(x > 0.1 X < 0.5). c) The conditional probability density function fx(x = 0.6 X > 0.5). [10 Marks [5 Marks [15 Marks]
3,40 A random variable X has probability density function fx(x) = 1 0<x< 1. Find the probability density function of Y = 4x3 - 2.
5. Suppose that the probability distribution function (p.d.f.) of a random variable X is as follows: a-x3) for 0<x<1 o/w Sketch this p.d.f. and determine the values of the following probabilities: f(x) =
2. A random variable has a probability density function given by: Bmx-(B+1) x20 x<m fx(x)= 10 where m>0 and B > 2. Let m and ß be constants; answer the questions in terms of m and B. (a) Find the cumulative distribution function (cdf) Fx(x) of this random variable; (b) Find the mean of X; (c) Find E[X']; and (d) Find the variance of X. [12 points]
Problem 5. The joint density of X and Y is given by e" (z+y) fx.-otherwise. İf 0 < x < oo, 0 < y < 00, Consider the random variable Z-; a) Find the cumulative distribution function of Z b) What is the probability density function of Z?
7. A random process may be modeled as a random variable with the probability density function fx(x) fx(x) fx(x) x < 1/2 x 1 /2 = 2-4x = 4x-2 = 0 (a) Show that Jdfx(r)dr = 1 (b) Find the CDF, Fx() (c) Find the probability that X is between 1/3 and 2/3. (d) Find the expected value of X. (e) Find the mode of X (f) Find the median of X. (g) Find the variance of X and the...
7. Let X be a random variable with distribution function Fx. Let a < b. Consider the following 'truncated' random variable Y: if X < a, if X > b. (a) Find the distribution function of Y in terms of Fx. (It will be a good additional exercise to sketch FY though you don't have to hand it in.) (b) Evaluate the limit lim FY (y) b-00
2x 0<x<1 Let X be a continuous random variable with probability density function f(x)= To else The cumulative distribution function is F(x). Find EX.
2.5.6. The probability density function of a random variable X is given by f(x) 0, otherwise. (a) Find c (b) Find the distribution function Fx) (c) Compute P(l <X<3)