3. Draw the cumulative distribution functions of the following distributions:(a) A continuous random variable chosen uniformly from the interval [1,6].(b) A continuous random variable chosen uniformly from the union [1,2] U [3, 4] U [5,6].(c) A discrete random variable chosen uniformly from the set {1, 2, 3, 4, 5, 6}.

Let X be a continuous uniform distribution Ua,b, then the probability density function of X is given…

Answered: Draw the cumulative distribution…

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...

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The continuous random variable X has a U (-6, 6) distribution, i.e. uniformly distributed between the points a = -6 and b = 6. What is the mean of X, and what is P (-3 < X < 1)?
let X(n,p) denotes binomial random variable with parameters n and p. Show that X(n+1,p) >st ( greater than equal ) X(n,p)
Prove a formula for the moment-generating function (mgf) of Y in terms of the mgf's of X₁ and N. For any random variable Z, the mgf is defined as oz(s) = E[e³Z], where s € IR and whenever the expectation is defined.
3. Draw the cumulative distribution functions of the following distributions: (a) A continuous random variable chosen uniformly from the interval [1,6]. (b) A continuous random variable chosen uniformly from the union [1,2] U [3, 4] U [5, 6]. (c) A discrete random variable chosen uniformly from the set {1, 2, 3, 4, 5, 6}.
5 Both K and M has standard normal distribution and they are independent random variables. Z=K+M Find E[K(K^2+M)^(1/4)].
b) calculate var(x) Give your answer to four decimal places.
2. We have two fair dice, one red and one blue. When we roll them together, the outcome can be shown as an order pair, (R, B) where R and B are numbers from the red and the blue die, respectively. Let X be a random variable defined by X(R, B) = R - B where R and B are numbers from red and blue dice, respectively. (a) What is the probability mass function for the random variable? Show that as a table.
Let (Ω, Pr) be a probability space, and let X and Y be two independent random variables that are positive and have non-zero variance. (a) Prove that X^2 and Y are independent. Note that by symmetry, it will also follow that Y^2 and X are independent. (b) Use the result from part (a) to show that the random variables W = X + Y and Z = XY are positively correlated (i.e. Cov(W, Z) > 0).
Let X(1), X(2), ..., X(n) be the order statistics of a set of n independent uniform (0, 1) random variables.Find the conditional distribution of X(n) given that X(1) = s(1), X(2) = s(2), ..., X(n-1)=s(n-1).
Let X1 and X2 be two independent and identically distributed random variables with N(µ, 1) distribution. Calculate the conditional distribution X1 | {X1 + X2 = s}.
Let X₁, X₂, ..., X5 be a random sample from N(1,4) and let Y₁, Y₂, Y3 be an independent random sample from N(3, 4). Find P > 1) using a proper F distribution, where S and S are sample variances for the two random samples respectively. In R/RStudio, the function of can be used for compute the probabilities based on F distribution.
A random point (X,Y) is distributed uniformly on the square with vertices (1,1), on the square. (1,-1), (-1, 1), and (-1,-1). That is, the joint pdf is f(x,y) Determine the probabilities of the following events. (a) x² + y² 0 (c) |X+Y| < 2

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