Page 81 - Probability and Statistical Inference

58 1. Notions of Probability

Find the value of c. Find the df F(x) and plot it. Does F(x) have any points of
discontinuity? Find the set of points, if nonempty, where F(x) is not differen-
tiable. Calculate P(– 1.5 < X ≤ 1.8). Find the median of this distribution.
1.6.4 (Example 1.6.5 Continued) Consider the function f(x) defined as
follows:

3
2
Let us denote the set A = {1, 1/2, 1/2 , 1/2 , ...}. Show that
(i) the function f(x) is a genuine pdf;
(ii) the associated df F(x) is continuous at all points x ∈ A;
(iii) Find the set of points, if nonempty, where F(x) is not differen-
tiable.
1.6.5 Along the lines of the construction of the specific pdf f(x) given in the
Exercise 1.6.4, examine how one can find other examples of pdf’s so that the
associated df’s are non-differentiable at countably infinite number of points.
1.6.6 Is f(x) = x I(1 < x < ∞) a genuine pdf? If so, find P{2 < X ≤ 3},
–2
P{|X – 1| ≤ .5} where X is the associated random variable.
1.6.7 Consider a random variable X having the pdf f(x) = c(x – 2x + x )I(0
2
3
< x < 1) where c is a positive number and I(.) is the indicator function. Find c
first and then evaluate P(X > .3). Find the median of this distribution.
1.6.8 Suppose that a random variable X has the Rayleigh distribution with

where θ(> 0) is referred to as a parameter. First show directly that f(x) is
indeed a pdf. Then derive the expression of the df explicitly. Find the median
of this distribution. {Hint: Try substitution u = x /θ during the integration.}
2
1.6.9 Suppose that a random variable X has the Weibull distribution with

where α(> 0) and β(> 0) are referred to as parameters. First show directly
that f(x) is indeed a pdf. Then derive the expression of the df explicitly. Find
β
the median of this distribution. {Hint: Try the substitution v = x /α during the
integration.}