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• ### 1. 20 points Let X be a random variable with the following probability density function: f(x)--e+1"...

1. 20 points Let X be a random variable with the following probability density function: f(x)--e+1" with ? > 0, ? > 0, constants x > ?, (a) 5 points Find the value of constant c that makes f(x) a valid probability mass function. (b) 5 points Find the cumulative distribution function (CDF) of X.

• ### 2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given...

2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)

• ### Suppose that a random variable X has a (probability) density function given by 52e-2, for x...

Suppose that a random variable X has a (probability) density function given by 52e-2, for x > 0; f(x) = 0, otherwise, (i) Calculate the moment generating function of X. [6 marks] (ii) Calculate E(X) and E(X²). [6 marks] (iii) Calculate E(ex/2), E(ex) and E(C3x), if they exist. [3 marks] (iv) Based on an independent random sample X = {X1, X2, ..., Xn} from the dis- tribution of X, provide a consistent estimator for 0 = E(esin(\)), where sin() is...

• ### PHYS1047 a) Given a random variable x, with a continuous probability distribution function fx) 4 marks...

PHYS1047 a) Given a random variable x, with a continuous probability distribution function fx) 4 marks b) The life expectancy (in days) of a mechanical system has a probability density write down equations for the cumulative distribution C(x) and the survival distribution Px). State a relationship between them. function f(x)=1/x, for x21, and f(x)=0 for x <1. i Find the probability that the system lasts between 0 and I day.2 marks i) Find the probability that the system lasts between...

• ### For a continuous random variable X with the following probability density function (PDF): fX(x) = (...

For a continuous random variable X with the following probability density function (PDF): fX(x) = ( 0.25 if 0 ≤ x ≤ 4, 0 otherwise. (a) Sketch-out the function and confirm it’s a valid PDF. (5 points) (b) Find the CDF of X and sketch it out. (5 points) (c) Find P [ 0.5 < X ≤ 1.5 ]. (5 points)

• ### 4. (30pts) A continuous random variable X has the probability density function: hx - 1 sx...

4. (30pts) A continuous random variable X has the probability density function: hx - 1 sx 32 f(x) =Jo-hx 2 x 3 0 x >3 which ean bo graphed as f(x) 1 2 a) Find h which makes f(x) a valid probability density function b) Find the expected value E(X) of the probability density function f(x) c) Find the cumulative distribution function F(x). Show all you work

• ### 4. The random variable X has probability density function f(x) given by f(x) = { k(2-...

4. The random variable X has probability density function f(x) given by f(x) = { k(2- T L k(2 - x) if 0 sxs 2 0 otherwise Determine i. the value of k. ii. P(0.7 sX s 1.2) iii. the 90th percentile of X.

• ### 1. Suppose the random variable X has the following probability density function: Problem Set: 1. Suppose...

1. Suppose the random variable X has the following probability density function: Problem Set: 1. Suppose the random variable X has the following probability density function: p(x) = fcx 0sxs2 10 otherwise. ] Note this probability density function is also of the form of an unknown parameter c. (a) Determine the value of c that makes this a valid probability density function. (b) Determine the expected value of X, E[X]. (c) Determine the variance of X, V(X).

• ### Q1: Suppose the probability density function of the magnitude X of a bridge (in newtons) is...

Q1: Suppose the probability density function of the magnitude X of a bridge (in newtons) is given by fx)-[e(1+3) sxs2 otherwise (a) Find the value of c. (b) Find the mean and variance (c) Find P(1 <x<2.25) (d) Find the cumulative distribution function.

• ### The density function of X is given by f(x) = a + bx2 if 0 ?...

The density function of X is given by f(x) = a + bx2 if 0 ? x ? 1 0 otherwise. Suppose also that you are told that E(X) = 3/5. (a) Find a and b. (b) Determine the cdf, F(x), explicilty. Problem 4. The density function of X is given by f(z) = 0 otherwise. Suppose also that you are told that E(X-3/5. (a) Find a and b. b) Determine the cdf, F(r