Section 3.1

Probability mass functions

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Problem 3.1.1

For what values of the constant CC do the following define mass functions on the positive integers 1,2,…1,2, \ldots?

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(a)

Geometric: f(x)=C2−xf(x) = C 2^{-x}.

(b)

Logarithmic: f(x)=C2−x/xf(x) = C 2^{-x} / x.

(c)

Inverse square: f(x)=Cx−2f(x) = C x^{-2}.

(d)

'Modified' Poisson: f(x)=C2x/xf(x) = C 2^{x} / x !.

Problem 3.1.2

For a random variable XX having each of the four mass functions of Exercise (3.1.1), find P(X>1)\mathbb {P}\left(X > 1\right), the most probable value of XX, and the probability that XX is even, when the mass function is:

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(a)

Geometric (Exercise 3.1.1(a)): f(x)=C2−xf(x) = C 2^{-x}.

(b)

Logarithmic (Exercise 3.1.1(b)): f(x)=C2−x/xf(x) = C 2^{-x} / x.

(c)

Inverse square (Exercise 3.1.1(c)): f(x)=Cx−2f(x) = C x^{-2}.

(d)

'Modified' Poisson (Exercise 3.1.1(d)): f(x)=C2x/x!f(x) = C 2^{x} / x!.

Problem 3.1.3

We toss nn coins, and each one shows heads with probability pp, independently of each of the others. Each coin which shows heads is tossed again. What is the mass function of the number of heads resulting from the second round of tosses?

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Problem 3.1.4

Let SkS_{k} be the set of positive integers whose base-10 expansion contains exactly kk elements (so that, for example, 1024∈S41024 \in S_{4}). A fair coin is tossed until the first head appears, and we write TT for the number of tosses required. We pick a random element, NN say, from STS_{T}, each such element having equal probability. What is the mass function of NN?

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Problem 3.1.5
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(a)

Show that, if XX is a binomial or Poisson random variable, then the mass function f(k)=P(X=k)f(k) = \mathbb {P}\left(X = k\right) has the property that f(k−1)f(k+1)≤f(k)2f(k-1) f(k+1) \leq f(k)^{2}.

(b)

Show that, if f(k)=90/(πk)4,k≥1f(k) = 90 /(\pi k)^{4}, k \geq 1, then f(k−1)f(k+1)≥f(k)2f(k-1) f(k+1) \geq f(k)^{2}.

(c)

Find a mass function ff such that f(k)2=f(k−1)f(k+1),k≥1f(k)^{2} = f(k-1) f(k+1), k \geq 1.