Question

2. Solve Each (a) How many ways can the letters in the word SLUMGULLION be arranged so that the three L's precede all the other consonants? (b) Give an example of a discrete random variable where expectation is infinite. Give its PDF explicitly and show that expectation is infinite. (c) Show for a continuous random variable X the limit $\lim_{x \to \infty} f_X(x)$ is not neces- sarily 0. Hint. Consider a function $g(x)$ such that $g(x) = a_k$ when $x \in [c_k, d_k]$ and $g(x) = 0$ outside of the intervals $[c_k, d_k]$. Now choose an appropriate se- quence of numbers ${a_k}$ and a sequence of intervals ${[b_k, c_k]}$ so that $g(x)$ is a PDF.

          2. Solve Each
(a) How many ways can the letters in the word SLUMGULLION be arranged so
that the three L's precede all the other consonants?
(b) Give an example of a discrete random variable where expectation is infinite.
Give its PDF explicitly and show that expectation is infinite.
(c) Show for a continuous random variable X the limit $\lim_{x \to \infty} f_X(x)$ is not neces-
sarily 0. Hint. Consider a function $g(x)$ such that $g(x) = a_k$ when $x \in [c_k, d_k]$
and $g(x) = 0$ outside of the intervals $[c_k, d_k]$. Now choose an appropriate se-
quence of numbers ${a_k}$ and a sequence of intervals ${[b_k, c_k]}$ so that $g(x)$ is a
PDF.

2. Solve Each
(a) How many ways can the letters in the word SLUMGULLION be arranged so
that the three L's precede all the other consonants?
(b) Give an example of a discrete random variable where expectation is infinite.
Give its PDF explicitly and show that expectation is infinite.
(c) Show for a continuous random variable X the limit limx →∞ fX(x) is not neces-
sarily 0. Hint. Consider a function g(x) such that g(x) = ak when x ∈ [ck, dk]
and g(x) = 0 outside of the intervals [ck, dk]. Now choose an appropriate se-
quence of numbers ak and a sequence of intervals [bk, ck] so that g(x) is a
PDF.

Added by Travis D.

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