Download Problem set 4: Functions of random variables 1. (Uniform and

Problem set 4: Functions of random variables
1. (Uniform and triangular distributions) Let U = X +Y where X and Y are independent standard
uniform random variables. Find the pdf f (u) of the random variable U . Why do you think the
distribution of U is called triangular distribution?
Hint: Consider the following cases: a) u < 0; b) 0 ≤ u < 1; c) 1 < u ≤ 2; d) u > 2.
2. (Uniform and exponential distributions) The exponential distribution E(λ) with parameter λ > 0
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was defined in Problem 3.2. Let X ∼ U [0, 1] and U = − ln X λ . Show that U ∼ E(λ).
3. (Gamma distribution)∗ A random variable X ∈ R is said to have gamma distribution, with a scale
parameter λ > 0 and shape parameter α > 0, if its pdf has the form
f (x) =
λα α−1 −λx
x e ,
Γ(α)
(x > 0),
where Γ(α) is the classical gamma function of Calculus defined by
Z ∞
Γ(α) =
xα−1 e−x dx.
0
We will use the shorthand notation X ∼ G(λ, α).
(a) Verify that f (x) is indeed a pdf.
(b) Sketch a plot of f (x) for 1) α < 1; 2) α = 1; 3) α > 1.
(c) Let X ∼ G(λ, α) and Y ∼ G(λ, β) be two independent gamma random variables (α, β, λ > 0).
Find the pdf of U = X + Y .
R1
Hint: 0 xα−1 (1 − x)β−1 dx = Γ(α)Γ(β)
.
Γ(α+β)
4. (Sums of exponential random variables) Let X1 , ..., Xn be n independent exponential E(λ) random variables. What is the distribution of X1 + X2 + · · · + Xn ?
Hint: G(λ, 1) = E(λ). Mathematical induction.
5. (Chi-squared distribution) Let X ∼ N (0, 1) be a standard normal random variable. It was shown
in ¡the class
that X 2 has chi-squared distribution χ2 (1), with 1 degree of freedom. Note that χ2 (1) ≡
¢
1 1
G 2, 2 .
(a) Let Y = σX. Show that Y 2 also has a gamma distribution and find its parameters.
(b) Let X1 , ..., Xn be n independent standard normal random variables N (0, 1). Find the pdf of
X12 + X22 + · · · + Xn2 . This distribution denoted χ2 (n) is called chi-squared distribution with n degrees
of freedom.
6. (Normal distribution) (a) Let X ∼ N (0, σ12 ) and Y ∼ N (0, σ22 ) be two independent normal variables. Show that X + Y ∼ N (0, σ12 + σ22 ).
(b) Let X ∼ N (µ1 , σ12 ) and Y ∼ N (µ2 , σ22 ) be two independent normal variables. Show that X + Y ∼
N (µ1 + µ2 , σ12 + σ22 ).
P
(c) P
Let X1 , P
X2 , ..., Xn be n independent random variables, Xi ∼ N (µi , σi2 ). Show that ni=1 Xi ∼
N ( ni=1 µi , ni=1 σi2 ).
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7. (Uniform and normal distributions – Box-Muller transform)∗ Let U and V be two independent random variables, each having standard uniform distribution on the interval [0, 1]. Consider new
variables
p
X = −2 log U cos(2πV ),
p
Y = −2 log U sin(2πV ).
a) Find the joint density of X and Y . (This method is often used in practice to generate standard
d
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normal r.v.’s from uniformly distributed r.v.’s.) Hint: dx
tan(−1) (x) = 1+x
2.
8. (Student’s t-distribution)∗ Let X be a standard normal random variable, and let U be an independent from X chi-squared random variable χ2 (n), with n degrees of freedom. Tee distribution of the
random variable
X
T =q
U
n
is called t distribution with n degrees of freedom, or sometimes Student’s t distribution. Find the
probability density of T .
9. (Fisher’s F -distribution) (a) Let U and V be two independent chi-squared random variables, χ2 (x)
and χ2 (s), with degrees of freedom r and s, respectively. The distribution of the random variable
F =
U/r
V /s
is called F distribution (or Fisher’s distribution), with r degrees of freedom in the nominator and s
degrees of freedom in the denominator. Find the probability density of F .
(b) Let Yi , 1 ≤ i ≤ n be n independent standard normal random variables. What is the distribution
of the random variable
Pp
Y 2 /p
Pn i=1 2 i
?
i=p+1 Yi /(n − p)
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