Page 82 - Probability and Statistical Inference

1. Notions of Probability 59

1.6.10 Consider the function f(x) = e (1 + x ) for x ∈ ℜ. Use (1.6.18) to
2
–x2
examine the monotonicity of f(x) in x.
1.6.11 Consider the function f(x) = exp{x – x – 1/2log(x)} for x > 0. Use
1/2
(1.6.18) to show that f(x) is increasing (decreasing) when |x + 1/2| < (>)3/4.
1/2
1.6.12 Consider the function f(x) = e (1 + x ) for x ∈ ℜ. Use (1.6.18) to
2
–|x|
examine the monotonicity of f(x) in x.
1.6.13 Use the method of integration by parts from (1.6.28) to evaluate

1.6.14 By the appropriate substitutions, express the following in the form
of a gamma integral as in (1.6.19). Then evaluate these integrals in terms of
the gamma functions.

1.6.15 Use Stirling’s approximation, namely that for large values of α, to
prove: for large positive integral values of n. {Hint: Observe
that n! =Γ(n+1) ~ √2 e -(n+1) (n+1) n+1/2 But, one can rewrite the last expression
as and then appeal to (1.6.13).}
1.7.1 Consider the random variable which has the following discrete uni-
form distribution:

Derive the explicit expression of the associated df F(x), x ∈ Β. Evaluate
and . {Hint: Use (1.6.11).}
1.7.2 The probability that a patient recovers from a stomach infection is 9.
Suppose that ten patients are known to have contracted this infection. Then
(i) what is the probability that exactly seven will recover?
(ii) what is the probability that at least five will recover?
(iii) what is the probability that at most seven will recover?
1.7.3 Suppose that a random variable X has the Binomial(n, p) distrib-