Let a random variable X has the following function: FX(x) = { 0 ; x < 0 (1/2)+(x/2) ; 0 ≤ x < 1 1 ; x ≥ 1. Verify that FX(x) is a CDF. Determine (i) the PDF of X, (ii) E[e^X], (iii) P[X = 0|X≤ 0.5].

Let a random variable X has the following function: FX(x) = { 0 ; x < 0 (1/2)+(x/2) ; 0 ≤ x < 1 1 ; x ≥ 1. Verify that FX(x) is a CDF. Determine (i) the PDF of X, (ii) E[e^X], (iii) P[X = 0|X≤ 0.5].

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Related questions

Q: satisfies F(-1) = 0.25, F(0) = 0.33, F(0.8) = 0.72, F(3.2) = 0.72, F(9.9) = 1. Find the value of…

A: We have given that the F(x) is a cumulative distribution function of a random variable X with…

Q: Exercise 5. Assume that we have observed the following values from a normal distribution with known…

A: we have following steps step-1)(it is given) null and alternative hypothesis H0:μ=0H1:μ=0…

Q: Let X denote the number of successes in a series of n independent Bernoulli trials with constant…

A: Given : Let X denote the number of successes in a series of n independentBernoulli trials with…

Q: (a) Find the expectation E (X) of X. E (X) = [] (b) Find the variance Var(X) of X. Varlx) = D

A: The following table shows the provided outputs of the discrete random variables, along with the…

Q: distributions exist: a a. fx(x) = -00 1 else (2x-3

Q: Let a random variable X has the CDF as; Fx(x) = 1 ; x ≥ 1 1/2+x ; 0 ≤ x < 1…

A: Random variable: Random variable is a real valued function that assign a real number to each outcome…

Q: The time for the first widget to be manufactured each morning is random between 1 and 3 seconds with…

A: Given density function: fx=1162x-32x+6, 0<x<6 Cumulative distribution function, F(X):…

Q: Let x be a random variable Gaussian with zero mean and variance 1. Find: a)The conditional pdf and…

A: To find the mean and Variance of positive half normal distribution

Q: For the optimization of the likelihood function, the computation of the log-likelihood L(8; y) = log…

A: In maximum likelihood estimation we maximize the likelihood function with respect to the parameters…

Q: 3. (a) What is the standard deviation of X? (b) Compute E[X²]. (c) Compute E[X(X - 1)].

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer three sub-parts at a…

Q: Let X be the total medical expenses (in 1000s of dollars) incurred by a particular individual during…

A: Since you have posted a question with multiple sub-parts, as per our guidelines we have provided…

Q: Consider a random variable X with the following pdf: fx (x) 2/9 1/9- 0 + 3 6 8 a) What is P(X ≤ 5)?…

A: PDf of random variable X is:

Q: Q.3 A function h(x) is given below. h(x)= (a) Test/check/prove whether h(x) is a valid CDF. (b)…

A: The given function:

Q: The values of x and their probabilities are listed below. Calculate the intermediate products to…

A: Given that: x P(x) x * p(x) 32 0.07 2.24 34 0.12 4.08 36 0.24 8.64 38 0.25 9.50 40 0.35…

Q: he life X (in years) of a voltage regulator of a car has the pdf f(x) = cx²e (²)³, x>0 a) Find c.…

A: Please note that as per our guidelines if multiple questions are posted we can answer the first…

Q: Let X denote the proportion of allotted time that a randomly selected student spends working on a…

Q: Suppose that the distribution of X is in the form f(x) = Cx4 for x in the interval [−1, 2]. (a)…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub- parts for…

Q: Suppose that X is a continuous random variable whose pdf satis fx(a) = 0 for all a 2. Prove that…

A: From the given information, the random variable X is a continuous random variable whose pdf…

Q: - x)² – 6 0< x < 2 otherwise, psitive. the cdf of X. nt Ber(0) random variables and le nerating…

A: C) Given data : sample size, n = 15 sample mean, x̄ = 46.75 sample standard…

Q: The continuous random variable X has the pdf f(x) = { (a) Find the cdf F of X. if x < 1 -2 (x-² if x…

A: Let X be a continuous random variable with the pdf given f(x),f(x)={0if x<1x−2if x≥1To find:a)…

Q: A random variable X has the pdf as f(x) = ax + b, for 0 < x< 1. The mean of X is E(X) = 2/3. (1)…

A: Given,f(x)=ax+b ; 0≤x≤1E(X)=23

Q: a) Find mean and variance of distribution by using M.G.F technique.(1 – p)* p

A: Note: Hey there! Thank you for the question. As you have posted multiple questions, we have helped…

Q: If random variable x is exponentially-distributed with mean a= 2, calculate p(X> 3)· Σ 2e-2x x=3 E…

A: From the provided information, Mean (λ) = 2 X~exp(2)

Q: The continuous random variable X has cumulative distribution function a. Find the PDF, fX(x) b.…

A: Here we have given the cumulative distribution function of continuous random variable X. We have…

Q: Q1 Let X be a non-negative discrete random variable. Prove that the expected value of X satisfies…

Q: Calculate the conditional expectation of Y when X E(YX2)

A: The objective is to calculate the conditional expectation of Y when X = 2.

Q: Pind the values of the constants a' and 'b' such that the CDF Кx (х) %3D 1- а.е xb] и (х) is a valid…

Q: Let X(1) 0. a) Show that X4) and X(6) – X() are independent for any s>r. b) Find the distribution of…

Q: Check whether the mean and the variance of the following distributions exist: a a. fx(x) = -00 < x <…

A: (a) it is the pdf of Cauchy distribution e.g X~C(a,1) Since know that mean and variance doesn't…

Q: f (x) = c/x3 for 1 < x < ∞; Find the constant c so that f (x) is a pdf of some random variable X,…

A: Given the functionwant to find the value of c such that the given function will be pdf. also wants…

Q: The instantaneous amplitude X of a speech signal is real-valued random variable described below:…

A: d) From the given information, Consider, the valid condition for f(x):

Q: A random variable X has a pdf [c(1-x*) -1<x <1 fx(x) otherwise (i) Find C. (ii) Find P[|x|<1/2].

Q: Let X be a continuous random variable with pdf f(x) = x² for -1 < x < c. Find the value of c.

A: From the given information, the pdf of the random variable X is,

Q: C1. Let X be a continuous random variable with PDF f(æ) = (2 – æ) for -1 1)? (c) Calculate the…

Q: A number X is randomly selected from the interval [10, 15]. The PDF of X is 1 15 - 10 0 1. Find the…

A: The pdf of a random variable X is given by: fXx=115-10 if x∈10, 15 0 Otherwise=15…

Q: f (x) = c/x3 for 1 < x < ∞; Find the constant c so that f (x) is a pdf of some random variable X,…

A: Let X be a continuous random variable and it is given thatf(x)=cx3 ;1<x<∞To find:Constant…

Q: Let P=f(t)=600(1.033)t be the population of a community in year t. Evaluate f(10)

Question

Let a random variable X has the following function:
FX(x) = {

0 ; x < 0
(1/2)+(x/2) ; 0 ≤ x < 1
1 ; x ≥ 1.

Verify that FX(x) is a CDF. Determine (i) the PDF of X, (ii) E[e^X], (iii) P[X = 0|X≤ 0.5].

Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Part a

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Multivariate Distributions and Functions of Random Variables$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.

Similar questions

The pdf of the time to failure of an electronic component in a copier (in hours) is f (x) = [exp (-x/3100)]/3100 for x > 0 and f (x) = 0 for x ≤ 0. Determine the probability that:(a) A component lasts more than 1180 hours before failure.(b) A component fails in the interval from 1180 to 2010 hours.(c) A component fails before 3010 hours.(d) Determine the number of hours at which 11% of all components have failed.(e) Determine the mean.Round your answers to three decimal places (e.g. 98.765).
Let X be a continuous random variable with PDF fx(x)= = { 10 1/8 1≤x≤9, otherwise. Let Y = h(X) = 1/√X. (a) Find E[X] and Var[X]. (b) Find h(E[X]) and E[h(X)]. Which one is greater? (c) Find E[Y] and Var[Y].
3 Let X be the number of siblings of WSU students. The pdf of X is: f(x) = (5-x)/15 for x = 0, 1, 2, 3, 4 What is the probability that a randomly selected student has 3 or more siblings if it is known that they are not an only child X 0.133 X 0.2 / 0.3 O X 0.667 X 0.933
A random process X(t) is applied as input to a system whose impulse response is h(t) 3u(t)t² exp (-8t). If E[X(t)] = 2, what is the mean value of the system response y(t). - CS Scanned with CamScanner
f (x) = c/x3/4 for 0 < x < 1; Find the constant c so that f (x) is apdf of some random variable X, and then find the cdf, F (x) = P(X ≤ x). Sketch graphsof the pdf f (x) and the cdf F (x), and find the mean μ and variance σ2.
Consider that a pdf of a random variable X is 1 -25x53 fx (x) ={K otherwise and another random variable Y = 2X. Then find (a) value K, (b) E[X], (c) E[Y] and ( d) EΧΥ.
Let X be a continuous random variable with cdf F(x). Show that E(I(X < x)) = F(r) where I is the indicator function (1 if X
X is a random variable that has the following PDF function: fx(x) = 0.28(x + 2) + 0.38(x + 1) + 0.358(x) + 0.158(x – 1) %3D For y = x2, find: 1- fy) 2- Fy(y) 3- E[y] 4- o
If E[X] = 1 and Var(X) = 5, use definition of variance and properties of expectation to find (a) E[(2 + X)^2] (b) V ar(aX) for any constant a. (c) V ar(X + b) for any constant b. (d) V ar(4 + 3X)