Page 79 - Probability and Statistical Inference

56 1. Notions of Probability

where p ∈ (0, .3). Is it possible to determine p uniquely? With a fixed but
arbitrary p ∈ (0, .3), find P{|X – .5| > 2}, P{X < 4} and P{|X – .3| = 1.7}.
2

1.5.3 Suppose that a random variable X has the following pmf:
X values: –4 –2 1 3 6
Probabilities: .3– p 2 p 2 .1 2p .4
where p ≤ .3. Is it possible to determine p uniquely?
2
1.5.4 (Example 1.5.2 Continued) Use the form of the df F(x) from the
Example 1.5.2 and perform the types of calculations carried out in (1.5.10).
1.5.5 A large envelope has twenty cards of same size. The number two is
written on ten cards, the number four is written on six cards, and the number
five is written on the remaining four cards. The cards are mixed inside the
envelope and we go in to take out one card at random. Let X be the number
written on the selected card.
(i) Derive the pmf f(x) of the random variable X;
(ii) Derive the df F(x) of the random variable X;
(iii) Plot the df F(x) and check its points of discontinuities;
(iv) Perform the types of calculations carried out in (1.5.10).
1.5.6 Prove the Theorem 1.5.1.
1.5.7 An urn contains m red and n blue marbles of equal size and weight.
The marbles are mixed inside the urn and then the experimenter selects four
marbles at random. Suppose that the random variable X denotes the number of
red marbles selected. Derive the pmf of X when initially the respective number
of marbles inside the urn are given as follows:
(i) m = 5, n = 3;
(ii) m = 3, n = 5;
(iii) m = 4, n = 2.
In each case, watch carefully the set of possible values of the random
variable X wherever the pmf is positive. Next write down the pmf of X when
m and n are arbitrary. This situation is referred to as sampling without re-
placement and the corresponding distribution of X is called the Hypergeomet-
ric distribution.
1.5.8 A fair die is rolled n times independently while we count the number
of times (X) the die lands on a three or six. Derive the pmf f(x) and the df F(x)
of the random variable X. Evaluate P(X < 3) and P(|X – 1| > 1).
1.6.1 Let c be a positive constant and consider the pdf of a random