If X is a random variable with CDF F(x) = e-e(µ-x)/β, where β > 0 and...

If X is a random variable with CDF F(x) = e^-e(µ-x)/β, where β > 0 and -∞ < µ, x < ∞; calculate the median of X. Also, obtain the PDF of X.

Expert Solution

orchestra answered 2 years ago

Let X be a random variable with CDF F(x) = e-e(µ-x)/β, where β > 0 and...

Let X be a random variable with CDF F(x) = e-e(µ-x)/β, where β > 0 and -∞ < µ, x < ∞. 1. What is the median of X? 2. Obtain the PDF of X. Use R to plot, in the range -10<x<30, the pdf for µ = 2, β = 5. 3. Draw a random sample of size 1000 from f(x) for µ = 2, β = 5 and draw a histogram of the values in the random sample...

Does a continuous random variable X exist with E(X − a) = 0, where a is...

Does a continuous random variable X exist with E(X − a) = 0, where a is the day of your birthdate? If yes, give an example for its probability density function. If no, give an explanation. (b) Does a continuous random variable Y exist with E (Y − b) 2 = 0, where b is the month of your birthdate? If yes, give an example for its probability density function. If no, give an explanation.

Suppose that random variable X 0 = (X1, X2) is such that E[X 0 ] =...

Suppose that random variable X 0 = (X1, X2) is such that E[X 0 ] = (µ1, µ2) and var[X] = σ11 σ12 σ12 σ22 . (a matrix) (i) Let Y = a + bX1 + cX2. Obtain an expression for the mean and variance of Y . (ii) Let Y = a + BX where a' = (a1, a2) B = b11 b12 0 b22 (a matrix). Obtain an expression for the mean and variance of Y . (ii)...

The random variable X has a continuous distribution with density f, where f(x) ={x/2−5i f10≤x≤12 ,0...

The random variable X has a continuous distribution with density f, where f(x) ={x/2−5i f10≤x≤12 ,0 otherwise. (a) Determine the cumulative distribution function of X.(1p) (b) Calculate the mean of X.(1p) (c) Calculate the mode of X(point where density attains its maximum) (d) Calculate the median of X, i.e. a number m such that P(X≤m) = 1/2 (e) Calculate the mean of the random variable Y= 12−X (f) Calculate P(X^2<121)

Let f(x) = a(e-2x – e-6x), for x ≥ 0, and f(x)=0 elsewhere. a) Find a...

Let f(x) = a(e-2x – e-6x), for x ≥ 0, and f(x)=0 elsewhere. a) Find a so that f(x) is a probability density function b)What is P(X<=1)

The following density function describes a random variable X. f(x)= (x/64) if 0<x<8 and f(x) =...

The following density function describes a random variable X. f(x)= (x/64) if 0<x<8 and f(x) = (16-x)/64 if 8<x<16 A. Find the probability that X lies between 2 and 6. B. Find the probability that X lies between 5 and 12. C. Find the probability that X is less than 11. D. Find the probability that X is greater than 4.

Let X be a exponential random variable with pdf f(x) = λe−λx for x > 0,...

Let X be a exponential random variable with pdf f(x) = λe−λx for x > 0, and cumulative distribution function F(x). (a) Show that F(x) = 1−e −λx for x > 0, and show that this function satisfies the requirements of a cdf (state what these are, and show that they are met). [4 marks] (b) Draw f(x) and F(x) in separate graphs. Define, and identify F(x) in the graph of f(x), and vice versa. [Hint: write the mathematical relationships,...

Let X be a uniform random variable with pdf f(x) = λe−λx for x > 0,...

Let X be a uniform random variable with pdf f(x) = λe−λx for x > 0, and cumulative distribution function F(x). (a) Show that F(x) = 1−e −λx for x > 0, and show that this function satisfies the requirements of a cdf (state what these are, and show that they are met). [4 marks] (b) Draw f(x) and F(x) in separate graphs. Define, and identify F(x) in the graph of f(x), and vice versa. [Hint: write the mathematical relationships,...

Considering X a continuous random variable defined by the distribution function f(x) = 0 if x<1...

Considering X a continuous random variable defined by the distribution function f(x) = 0 if x<1 -k +k/x if 1<= x < 2 1 if 2<x Find k.

Let X be a continuous random variable with pdf: f(x) = ax^2 − 2ax, 0 ≤...

Let X be a continuous random variable with pdf: f(x) = ax^2 − 2ax, 0 ≤ x ≤ 2 (a) What should a be in order for this to be a legitimate p.d.f? (b) What is the distribution function (c.d.f.) for X? (c) What is Pr(0 ≤ X < 1)? Pr(X > 0.5)? Pr(X > 3)? (d) What is the 90th percentile value of this distribution? (Note: If you do this problem correctly, you will end up with a cubic...

Question