Let X1, X2,..., Xnbe independent and identically distributed exponential random variables with parameter λ . a)...

Let X₁, X₂,..., X_nbe independent and identically distributed exponential random variables with parameter λ .

a) Compute P{max(X₁, X₂,..., X_n) ≤ x} and find the pdf of Y = max(X₁, X₂,..., X_n).
b) Compute P{min(X₁, X₂,..., X_n) ≤ x} and find the pdf of Z = min(X₁, X₂,..., X_n).
c) Compute E(Y) and E(Z).

Solutions

Expert Solution

milcah answered 10 months ago

Next > < Previous

Related Solutions

Let X1, X2, . . . be a sequence of independent and identically distributed random variables...

Let X1, X2, . . . be a sequence of independent and identically distributed random variables where the distribution is given by the so-called zero-truncated Poisson distribution with probability mass function; P(X = x) = λx/ (x!(eλ − 1)), x = 1, 2, 3... Let N ∼ Binomial(n, 1−e^−λ ) be another random variable that is independent of the Xi ’s. 1) Show that Y = X1 +X2 + ... + XN has a Poisson distribution with mean nλ.

Let X1,...,Xn be exponentially distributed independent random variables with parameter λ. (a) Find the pdf of...

Let X1,...,Xn be exponentially distributed independent random variables with parameter λ. (a) Find the pdf of Yn= max{X1,...,Xn}. (b) Find E[Yn]. (c) Find the median of Yn. (d) What is the mean for n= 1, n= 2, n= 3? What happens as n→∞? Explain why.

Let X1 and X2 be independent identically distributed random variables with pmf p(0) = 1/4, p(1)...

Let X1 and X2 be independent identically distributed random variables with pmf p(0) = 1/4, p(1) = 1/2, p(2) = 1/4 (a) What is the probability mass function (pmf) of X1 + X2? (b) What is the probability mass function (pmf) of X(2) = max{X1, X2}? (c) What is the MGF of X1? (d) What is the MGF of X1 + X2

Let X1 and X2 be independent identically distributed random variables with pmf p(0) = 1/4, p(1)...

2. Let X1, X2, . . . , Xn be independent, uniformly distributed random variables on...

2. Let X1, X2, . . . , Xn be independent, uniformly distributed random variables on the interval [0, θ]. (a) Find the pdf of X(j) , the j th order statistic. (b) Use the result from (a) to find E(X(j)). (c) Use the result from (b) to find E(X(j)−X(j−1)), the mean difference between two successive order statistics. (d) Suppose that n = 10, and X1, . . . , X10 represents the waiting times that the n = 10...

Suppose X1, X2, . . ., Xn are iid Poisson random variables with parameter λ. (a)...

Suppose X1, X2, . . ., Xn are iid Poisson random variables with parameter λ. (a) Find the MVUE for λ. (b) Find the MVUE for λ 2

Let ?1 , ?2 , ... , ?? be independent, identically distributed random variables with p.d.f....

Let ?1 , ?2 , ... , ?? be independent, identically distributed random variables with p.d.f. ?(?) = ???−1, 0 ≤ ? ≤ 1 . c) Show that the maximum likelihood estimator for ? is biased, and find a function of the mle that is unbiased. (Hint: Show that the random variable −ln (??) is exponential, the sum of exponentials is Gamma, and the mean of 1/X for a gamma with parameters ? and ? is 1⁄(?(? − 1)).) d)...

Suppose that X1, X2, X3, X4 is a simple random (independent and identically distributed) sample of...

Suppose that X1, X2, X3, X4 is a simple random (independent and identically distributed) sample of size 4 from a normal distribution with an unknown mean μ but a known variance 9. Suppose further that Y1, Y2, Y3, Y4, Y5 is another simple random sample (independent from X1, X2, X3, X4 from a normal distribution with the same mean μand variance 16. We estimate μ with U = (bar{X}+bar{Y})/2. where bar{X} = (X1 + X2 + X3 + X4)/4 bar{Y}...

Let X1,X2,… be a sequence of independent random variables, uniformly distributed on [0,1]. Define Nn to...

Let X1,X2,… be a sequence of independent random variables, uniformly distributed on [0,1]. Define Nn to be the smallest k such that X1+X2+⋯+Xn exceeds cn=n2+12n−−√, namely, Nn = min{k≥1:X1+X2+⋯+Xk>ck} Does the limit limn→∞P(Nn>n) exist? If yes, enter its numerical value. If not, enter −999. unanswered Submit You have used 1 of 3 attempts Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.

If X and Y are independent exponential random variables, each having parameter λ = 6, find...

If X and Y are independent exponential random variables, each having parameter λ = 6, find the joint density function of U = X + Y and V = e 2X. The required joint density function is of the form fU,V (u, v) = { g(u, v) u > h(v), v > a 0 otherwise (a) Enter the function g(u, v) into the answer box below. (b) Enter the function h(v) into the answer box below. (c) Enter the value...

Subjects