L.9) Hand calculation of expected values a) The pdf of a random variable X with the...
1. Suppose that X is continuous random variable with PDF f(x) and CDF F(x). (a) Prove that if f(x) > 0 only on a single (possible infinite) interval of the real numbers then F(x) is a strictly increasing function of x over that interval. [Hint: Try proof by contradiction]. (b) Under the conditions described in part (a), find and identify the distribution of Y = F(x). 2. Suppose now that X ~ Uniform(0, 1). For each of the distributions listed...
Problem 5. Let X be a continuous random variable with a 2-paameter exponential distribution with parameters α = 0.4 and xo = 0.45, ie, ;x 2 0.45 x 〈 0.45 f(x) = (2.5e-2.5 (-0.45) Variable Y is a function of X: a) Find the first order approximation for the expected value and variance of Y b) Find the probability density function (PDF) of Y. c) Find the expected value and variance of Y from its PDF
Problem 5. Let X...
Problem 2 (9 points) Consider a random variable X with pdf given by: (x) 0.06x +0.05 0x <5 4pts P (3.5 < X < 6.5)- Find Find Ex]- 5pts Problem 3 (9 points) Consider a random variable X with pdf given by: f(x) 0.06x +0.05 x -0, 1.5, 2, 4, 5 .ind P(3.5 <X <6.5)- 4pts 5pts Given that EN-3.46, find al
a) The pdf of a random variable X is (1-μ e 26 The generating function of X is t2 -2 Use what you see to write down the Fourier transform of pdf[x] b) What is the relation between The Fourier transform of pdf[x] and the characteristic function of X? c) If the pdfs of two random variables have the same Fourier transform, then must they have the same cumulative distribution function? L.14 The pdf of a random variable X is...
3. Consider a continuous random variable X with pdf given by 0, otherwise This is called the exponential distribution with parameter X. (a) Sketch the pdf and show that this is a true pdf by verifying that it integrates to 1 (b) Find P(X < 1) for λ (c) Find P(X > 1.7) for λ : 1
Problem #3. X is a random variable with an exponential distribution with rate 1 = 3 Thus the pdf of X is f(x) = le-ix for 0 < x where = 3. a) Using the f(x) above and the R integrate function calculate the expected value of X. b) Using the dexp function and the R integrate command calculate the expected value of X. c) Using the pexp function find the probability that .4 SX 5.7 d) Calculate the probability...
Calculation need to be done using R
Suppose X is a random variable taking on possible values 1,2,3 with respective probabilities.4, .5, and .1. Y is a random variable independent from X taking on possible values 2,3,4 with resp probabilities.3, .3, and .4. Use R to determine the following. a) Find the probability P(X*Y = 4) b) Find the expected value of X. c) Find the standard deviation of x. d) Find the expected value of Y. e) Find the...
4. Let X be a random variable with pdf f(x). Suppose that the mean of X is 2 and the variance of X is 5. It is easy to show that the pdf of Y = 0X is fo(y) = f(1/0) (You do not have to show this, but it's good practice.) Suppose the popula- tion has the distribution of foly) with 8 unknown. We take a random sample {Y}}=1 and compute the sample mean Y. (a) What is a...
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is exponential with parameter equal to r (and mean 1/r) Note: Some useful integrals, for λ > 0: ar (a) Find the joint PDF of X and Y (b) Find the marginal PDF of Y (c) Find the conditional PDF of X, given that Y 2. (d) Find the conditional expectation of X, given that Y 2 (e) Find the...
Consider an exponentially distributed random variable X with pdf f(x) = 2e−2x for x ≥ 0. Let Y = √X. a. Find the cdf for Y. b. Find the pdf for Y. c. Find E[Y]. If you want to skip a difficult integration by parts, make a substitution and look for a Gamma pdf. d. This Y is actually a commonly used continuous distribution. Can you name it and identify its parameters? e. Suppose that X is exponentially distributed with...