1. Let X have a normal distribution with parameters μ = 50 and σ2 = 144....

1. Let X have a normal distribution with parameters μ = 50 and σ2 =
144. Find the probability that X produces a value between 44 and 62. Use the
normal table A7 (be sure to show your work).

2. Let X ~ Exponential( λ ), for some fixed constant λ > 0. That is,
fX(x) = λ e-λx = λ exp( -λx ), x > 0, ( fX(x) = 0 otherwise)
(a) Create a transformed random variable Y = X1/3 (cube root of X). Using
either the CDF method or Jacobian method, show that the probability
density function of Y is given by: (10 pts)
fY(y) = 3λy^2e^(-λy3) , 0 < y < ∞.
(b) Verify that the CDF of Y is FY(y) = 1 - exp( -λy3 ), 0 < y < ∞. (5 pts)
(Hint: Use the CDF from (a) or integrate the PDF answer in (a)).
(c) If λ = 2, compute the median of Y (approximately). (5 pts)

Expert Solution

i

if you are refering to the table above then the values are in the last row, ideally the table below should be the real standard normal table( biometrica format)

orchestra answered 2 years ago

let the random variable x follow a normal distribution with μ = 50 and σ2 =...

let the random variable x follow a normal distribution with μ = 50 and σ2 = 64. a. find the probability that x is greater than 60. b. find the probability that x is greater than 35 and less than 62 . c. find the probability that x is less than 55. d. the probability is 0.2 that x is greater than what number? e. the probability is 0.05 that x is in the symmetric interval about the mean between...

et the random variable x follow a normal distribution with μ = 50 and σ2 =...

et the random variable x follow a normal distribution with μ = 50 and σ2 = 64. a. find the probability that x is greater than 60. b. find the probability that x is greater than 35 and less than 62 . c. find the probability that x is less than 55. d. the probability is 0.2 that x is greater than what number? e. the probability is 0.05 that x is in the symmetric interval about the mean between...

If X is a normal random variable with parameters σ2 = 36 and μ = 10,...

If X is a normal random variable with parameters σ2 = 36 and μ = 10, compute (a) P{X ≥ 5} . (b) P{X = 5}. (c) P{10>X≥5}. (d) P{X < 5}. (e) Find the y such that P{X > y} = 0.1.

Let X ∼ Normal(μ = 20, σ2 = 4). (a) Give the mgf MX of X....

Let X ∼ Normal(μ = 20, σ2 = 4). (a) Give the mgf MX of X. (b) Find the 0.10 quantile of X. (c) Find an interval within which X lies with probability 0.60. (d) Find the distribution of Y = 3X −10 by finding the mgf MY of Y

Let the random variable X follow a normal distribution with a mean of μ and a...

Let the random variable X follow a normal distribution with a mean of μ and a standard deviation of σ. Let 1 be the mean of a sample of 36 observations randomly chosen from this population, and 2 be the mean of a sample of 25 observations randomly chosen from the same population. a) How are 1 and 2 distributed? Write down the form of the density function and the corresponding parameters. b) Evaluate the statement: ?(?−0.2?< ?̅1 < ?+0.2?)<?(?−0.2?<...

2] Let x be a continuous random variable that has a normal distribution with μ =...

2] Let x be a continuous random variable that has a normal distribution with μ = 48 and σ = 8 . Assuming n N ≤ 0.05 , find the probability that the sample mean, x ¯ , for a random sample of 16 taken from this population will be between 49.64 and 52.60 . Round your answer to four decimal places.

Let x be a continuous random variable that has a normal distribution with μ = 60...

Let x be a continuous random variable that has a normal distribution with μ = 60 and σ = 12. Assuming n ≤ 0.05N, where n = sample size and N = population size, find the probability that the sample mean, x¯, for a random sample of 24 taken from this population will be between 54.91 and 61.79. Let x be a continuous random variable that has a normal distribution with μ = 60 and σ = 12. Assuming n...

Suppose X1, . . . , Xn is a random sample from the Normal(μ, σ2) distribution,...

Suppose X1, . . . , Xn is a random sample from the Normal(μ, σ2) distribution, where μ is unknown but σ2 is known, and it is of interest to test H0: μ = μ0 versus H1: μ ̸= μ0 for some value μ0. The R code below plots the power curve of the test Reject H0 iff |√n(X ̄n − μ0)/σ| > zα/2 for user-selected values of μ0, n, σ, and α. For a sequence of values of μ,...

Let Yi equal a normal distribution with mean β + βxi and variance σ2 = 1...

Let Yi equal a normal distribution with mean β + βxi and variance σ2 = 1 for i = 1; 2; :::; n. Further, suppose that the Yi are independent and that the xi are known values. Find the MLE of β 0 and β1.

Let X have a Poisson distribution with mean μ . Find E(X(X-1)) and use it to...

Let X have a Poisson distribution with mean μ . Find E(X(X-1)) and use it to prove that μ = σ 2 .

Question