372 CHAPTER 6 Inferential Statistics
- In Exercise 6 on page 362 you were asked essentially to investigate
the distribution of X 1 +X 2 whereX 1 and X 2 were independent
exponential random variables, each with mean μ = 1/λ. Given
that the density function of each isf(x) =λe−λxand given that the
sum has as density the convolution off with itself (see page 370),
compute this density. - In Exercise 8 on page 356 we showed that the density function
for the χ^2 random variable with one degree of freedom is f(x) =
√^1
2 π
x−^1 /^2 e−x/^2. Using the fact that the χ^2 with two degrees of
freedom is the sum of independent χ^2 random variables with one
degree of freedom, and given that the density function for the
sum of independent random variables is the convolution of the two
corresponding density functions, compute the density function for
the χ^2 random variable with two degrees of freedom. (See the
footnote on page 356.)- Letfbe anevenreal-valued function such that
∫∞
−∞f(x)dxexists.
Show thatf∗f is also an even real-valued function.- Consider the function defined by setting
f(x) =
x^2 if − 1 ≤x≤ 1 ,
0 otherwise.(a) Show thatf∗f(x) =
∫x+1
− 1 y(^2) (x−y) (^2) dy if − 1 ≤x≤ 0 ,
∫ 1
x− 1 y
(^2) (x−y) (^2) dy if 0≤x≤ 1.