Question

In: Statistics and Probability

In Example 6.33 we found the distribution of the sum of two i.i.d. exponential variables with...

In Example 6.33 we found the distribution of the sum of two i.i.d. exponential variables with parameter λ. Call the sum X. Let Y be a third independent exponential variable with parameter λ. Use the convolution formula 6.8 to find the sum of three independent exponential random variables by finding the distribution of X + Y.

Expert Solution

Answer:

Given Data

we found the distribution of the sum of two i.i.d. exponential variables with parameter λ.

Let Y be a third independent exponential variable with parameter λ.

x = sum of two expoential distribution.

distribution

and distribution

let pdf of x be f(x)

and pdf of y be g(y)

need to find pdf of z = x + y

so let pdf of z be h(z)

then by convolution formula:

now

, ,

,

so

Hence

,

so distribution

same way it can be proven that distribution

if each distribution.

orchestra answered 1 year ago

Suppose X1, ..., Xn are i.i.d. from an exponential distribution with mean θ. If we are...

Suppose X1, ..., Xn are i.i.d. from an exponential distribution with mean θ. If we are testing H0 : θ = θ0 vs Ha : θ > θ0. Suppose we reject H0 when ( X¯n/ θ0) > 1 + (1.645/ √n) (a) (10 points) Calculate the power function G(ζ). You may leave your answer in terms of the standard normal cdf Φ(x). (b) (5 points) Is this test consistent?

Suppose X1; : : : ; Xn is i.i.d Exponential distribution with density f(xjθ) = (1/θ)...

Suppose X1; : : : ; Xn is i.i.d Exponential distribution with density f(xjθ) = (1/θ) * e(-x/θ); 0 ≤ x < 1; θ > 0: (a) Find the UMVUE (the best unbiased estimator) of θ. (b) What is the Cramer-Rao lower bound of all unbiased estimator of all unbiased estimator of θ. Does the estimator from (a) attain the lower bound? Justify your answer. (c) What is the Cramer-Rao lower bound of all unbiased estimator of θ^2? 3 (d)...

distinguish between the exponential distribution and the poisson distribution

Suppose we wish to generate a sample from the exponential ($\beta$) distribution, and only have access...

Suppose we wish to generate a sample from the exponential ($\beta$) distribution, and only have access to a computer which generates numbers from the skew logistic distribution. It turns out that if $X$~SkewLogistic ($\beta$), then log(1+exp($-X$)) is exponential ($\beta$). Show that this is true and check by simulation that this transformation is correct.

R simulation: Let X1, . . . , Xn be i.i.d. random variables from a uniform...

R simulation: Let X1, . . . , Xn be i.i.d. random variables from a uniform distribution on [0, 2]. Generate and plot 10 paths of sample means from n = 1 to n = 40 in one figure for each case. Give some comments to empirically check the Law of Large Numbers. (a) When n is large, X1 + · · · + Xn/n converges to E[Xi]. (b) When n is large, X1^2+ · · · + Xn^2/n converges to...

(Exponential Distribution) The life, in years, of a certain type of electrical switch has an exponential...

(Exponential Distribution) The life, in years, of a certain type of electrical switch has an exponential distribution with an average life of ?? = 2 years. i) What is the probability that a given switch is still functioning after 5 years? ii) If 100 of these switches are installed in different systems, what is the probability that at most 30 fail during the first year?(also Binomial Distribution

Suppose that ?1, ?2, … ?? are i.i.d RV drawn from a normal distribution N(2,5) distribution...

Suppose that ?1, ?2, … ?? are i.i.d RV drawn from a normal distribution N(2,5) distribution . a. What is the mean and the variance of sample average where i.) n=10, ii) n=100, iii) n=1000.

Scatter diagrams allow us to see relationships between two variables. Cite an example of two variables...

Scatter diagrams allow us to see relationships between two variables. Cite an example of two variables that might have a strong positive relationship. What would that scatter diagram look like? Next, cite an example of two variables that have very little relationship and describe what that scatter diagram would look like. Last, tell us why illustrating data in this way is helpful. Can this be typed out? It's for a discussion board.

1. A product of two functions (exponential) 2. A quotient of two functions (logarithmic) 3. example...

1. A product of two functions (exponential) 2. A quotient of two functions (logarithmic) 3. example of a composite function 4. A sum of two functions (rational) 5. A difference of two function (rational or either trigonometric) for all the functuons you come up w please give the domin range x and y int ans local max and min.

Assume that the time between two consecutive accidents in a chemistry lab follows an exponential distribution...

Assume that the time between two consecutive accidents in a chemistry lab follows an exponential distribution with parameter λ. Starting to count from the day of the first accident (this will be day 0), there has been accidents on the 28th, 50th, 60th days. Compute the maximum likelihood estimate of λ. Explain the steps.