Let ?1, ?2, ?3 be 3 independent random variables with uniform distribution on [0, 1]. Let...

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).

Solutions

Expert Solution

orchestra answered 1 year ago

Next > < Previous

Related Solutions

2. Let ?1, ?2, ?3 be 3 independent random variables with uniform distribution on [0, 1]....

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 ?1 and ?2 be two independent random variables with uniform distribution on [0, 1]. 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 that says that the equation ?^2 −?1?...

Let ?1 and ?2 be two independent random variables with uniform distribution on [0, 1]. 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 X, Y be independent random variables with X ∼ Uniform([1, 5]) and Y ∼ Uniform([2,...

Let X, Y be independent random variables with X ∼ Uniform([1, 5]) and Y ∼ Uniform([2, 4]). a) FindP(X<Y). b) FindP(X<Y|Y>3) c) FindP(√Y<X<Y).

1. Let ?1, . . . ?? be ? independent random variables with normal distribution 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....

Let ? and ? be two independent uniform random variables such that ?∼????(0,1) and ?∼????(0,1). A)...

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 ??

Let ? ∈ {1, 2} and ? ∈ {3, 4} be independent random variables with PMF-s:...

Let ? ∈ {1, 2} and ? ∈ {3, 4} be independent random variables with PMF-s: ?x(1)= 1/2 ?x(2)= 1/2 ?Y(3)= 1/3 ?Y(4)= 2/3 (a) Write down the joint PMF (b) Calculate ?(? + ? ≤ 5) and ? (? − ? ≥ 2) (c) Calculate ?(?? ), ?(?2? ),Calculate E((X2+1)/(Y-2)) (d) Calculate the C??(?, ? ), C??(1 − ?, 3? + 2) and V??(2? − ? ) (e) Calculate C??(??, ?), C??(??, ? + ? ) and V?? (?/Y)

Let ? ∈ {1, 2} and ? ∈ {3, 4} be independent random variables with PMF-s:...

Let ? ∈ {1, 2} and ? ∈ {3, 4} be independent random variables with PMF-s: ??(1)= 1/2 ??(2)= 1/2 ??(3)= 1/3 ??(4)= 2/3 Answer the following questions (a) Write down the joint PMF (b) Calculate?(?+?≤5)and?(? −?≥2) 2 ?2+1 (c) Calculate ?(?? ), ?(? ? ), E ? −2 (d) Calculate the C??(?, ? ), C??(1 − ?, 3? + 2) and V??(2? − ? ) (?*) Calculate C??(??, ?), C??(??, ? + ? ) and V?? ?

Let X and Y be uniform random variables on [0, 1]. If X and Y are...

Let X and Y be uniform random variables on [0, 1]. If X and Y are independent, find the probability distribution function of X + Y

Subjects