Let ?1 and ?2 be two independent random variables with uniform
distribution on [0, 1].
1. Write down the joint cumulative distribution function and joint
probability
density function of ?1 + ?2 and ?1?2.
2. Write down the covariance between ?1 + ?2 and ?1?2.
3. Let ? be the largest magnitude (absolute value) of a root of the
equation
?^2 − ?1? + ?2 = 0. Let ? be the random event that says that
the
equation ?^2 −?1?...
Let ?1 and ?2 be two independent random variables with uniform
distribution on [0, 1].
1. Write down the joint cumulative distribution function and
joint probability density function of ?1 + ?2 and ?1?2.
2. Write down the covariance between ?1 + ?2 and ?1?2.
3. Let ? be the largest magnitude (absolute value) of a root of
the equation ? 2 − ?1? + ?2 = 0. Let ? be the random event which
says that the equation ?...
Let ?1, ?2, ?3 be 3 independent random variables with uniform
distribution on [0, 1]. Let ?? be the ?-th smallest among {?1, ?2,
?3}. Find the variance of ?2, and the covariance between the median
?2 and the sample mean ? = 1 3 (?1 + ?2 + ?3).
2. Let ?1, ?2, ?3 be 3 independent random variables with uniform
distribution on [0, 1]. Let ?? be the ?-th smallest among {?1, ?2,
?3}. Find the variance of ?2, and the covariance between the median
?2 and the sample mean ? = 1 3 (?1 + ?2 + ?3).
Let ? and ? be two independent uniform random
variables such that
?∼????(0,1) and
?∼????(0,1).
A) Using the convolution formula, find the pdf
??(?) of the random variable
?=?+?, and graph it.
B) What is the moment generating function of ??
1. Let ?1, . . . ?? be ? independent random variables with
normal distribution of expectation 0 and variance ? 2 . Let ?̂︁2 1
be the sample variance ??, ?̂︁2 2 be 1 ? ∑︀ ? ?2 ? . (1) Show that
the expectation of ?̂︁2 1 and ?̂︁2 2 are both ? 2 . In other words,
both are unbiased point estimates of ? 2 . (2) Write down the
p.d.f. of ?̂︁2 1 and ?̂︁2 2....
Problem 1 Let X1, X2, . . . , Xn be independent Uniform(0,1)
random variables. (
a) Compute the cdf of Y := min(X1, . . . , Xn).
(b) Use (a) to compute the pdf of Y .
(c) Find E(Y ).
Let X1, X2, . . . be iid random variables following a uniform
distribution on the interval [0, θ]. Show that max(X1, . . . , Xn)
→ θ in probability as n → ∞