-x/2 x > 0 (a) f(x) = otherwise Jc(1 – x²) -1 < x < 1. |0 (b) f(x) = otherwise 5 x > 5 (c) f(x) = {x2 x < 5

Q: Suppose X1 and X2 have joint density function { fx,,x2 ("1, #2) 0, O.W. E[X1]E[X2] Find the marginal…

A: GivenJoint density functionfX1,X2(x1,x2)=1πx12+x22≤10otherwise Let us say X1 range from -1 to 1,…

Q: 3. Determine the constant a so that f(x) is a valid density function. - f(x) = {a(3 — x²), a) 0.3333…

A: Ans#- Given, f(x) is valid density function           f(x) = a(3 - x2)-1 ≤x≤10otherwise…

Q: Let X and Y be independent and exponential with parameter lambda. Show that the density of X…

A: Given that X and Y be independent and exponential random variables with parameter lambda.

Q: Let X and Y have the joint density function 2, 0<x<y; 0<y<1, f(x,y)= %3D 0, Otherwise. Then the…

A:

Q: Let X and Y have the following joint probability density function: fxy(x,y)=1/(x²y²), for x>1, y>1…

A: The PDF of X and Y is given as:Also, the given transformation is:U=5XYandV=8X

Q: For some € (0, 1), define the density on the interval (0, 2) : 0x € (0,1], 0 V (1,2). (a) Verify…

A: Let X be a random variable, a real valued function f(x) will be a probability density function if1)…

Q: Let f (2) = #{1- ()*}. 1-()}, a – B 0. Find k so that fis a density.

A:

Q: Let (X,Y) have the joint density function f(x,y) = { (6/5)(1+x+y) for 0 s x, y < 1, x+ys1 otherwise…

A: The joint density function (X,Y) is fx,y=651+x+y    for  0≤x,y≤1,  x+y≤1          0…

Q: Suppose that X has density function fx (x) : 2e-2x for x > 0 (and fx (x) = 0 otherwise). Also…

A: # Given pdf of x :f(x)=2*e^-2x :x&gt;0  And conditional probability distribution of y/x is…

Q: For some 0 € (0, 1), define the density on the interval (0, 2): 0 V € (0,1], ƒ (x|0) = { 1-0 Vr…

A: The probability density function of random variable X is given as:

Q: What is the joint probability density function fy,Y, Oof Y1 = X1/X2 and Y, = X,?

A: It is given that  Y1 = X1/X2 and Y2 = X2 We will use Jacobian method of transformation to find the…

Q: - The spectral density of a process {X(t)}, real or complex, is a real function of w and non…

A:

Q: Find E(X)

A:

Q: Let X and Y have the following joint probability density function: fx,y(x,y) = 1 / (x² y²), for x>1,…

A: The PDF of X and Y is given as:Also, the given transformation is:U=9XYandV=10X

Q: Let X be exponential with parameter ß = 1. The CDF of X is F(x) = 1- e-* 1 Let U = What is the X'…

A:

Q: Let X and Y have the following joint probability density function: {xr(x,y) =1/ (x2 y2), forx>1, y>1…

A: 1. From the given information, fx,y=1x2y2, x&gt;1, y&gt;1 Let U=6XY, V=7X V=7XV7=X And…

Q: Let X have the probability density function (PDF) given by fx(x) = {x, 1½ x, 0, 0 <x < 2 otherwise…

A: The transformation of the random variable X with PDF as f(x) with respect to the function Y can be…

Q: Find the conditional density functions for the joint density function fxy(x, y) = 4xy e¯(x²+y²)…

A:

Q: It is estimated that 50% of chata are spam chats. Some software has been applied to filter these…

A: Solution

Q: .) Define the constant c ∈ ƒ so that the function f is a density function. In addition, define the…

A: Given information y=lnX  And f(x)=c/x; √e&lt;x&lt;e

Q: Use the Partition Theorem (or otherwise) to determine E (Y), the expected value of Y.

A: The partition theorem says that E(Y) = E[E(Y|X=x)]

Q: Using the transformation theorem, find the density joint of the random vector (Y1, Y2). 2. From the…

A:

Q: You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 the…

A: a) The probability model for the amount you win is:EventProbabilityWinnings61/6$100Not 65/6 *…

Q: Let and Y have the joint density function 2, 0<x<1; 0 <y<x, f(x,y)= 0, Otherwise. Calculate E(XY).

A:

Q: For any positive number k, verify that f(x) = 1/(2k³) x²e¬v/k, x2 0, is a density function.

A: For a density function, the total probability must be equal to 1.  ∫0∞f(x)dx=1

Q: Let X have probability mass function f(x)=x/6, x=1,2,3. Find E(Y) when Y=X2

A: For a discrete random variable X   E(X) = ∑xX f(x)E(X2) = ∑xX2 f(x)   Here, Let X have a probability…

Q: Let X and Y have the joint density function 0<x<1; 0<y<x, 2, f(x,y)= 0, Otherwise. Calculate E(XY).

A:

Q: Find the cumulative distribution function for Y if its density is f(y) = { 0, S 2(1 – 1/y²), 1 1.5).

A: Solution: From the given information, the probability density function of Y is

Q: Let Z have the following density functioI fzlz) =, 5sz57 0, otherwise. %3D 17 mi7

A: Probability Density function (PDF) is the probability function which is represented for the density…

Q: 4. Consider the density function 0.5 <x < 2.5 f(x) = }kvx, 0, %3D elsewhere. a) Evaluate k. b) Find…

A: a) The total area under the probability density function is one.…

Q: Suppose (y/) ~ Poisson(0). Find Jeffrey's prior density for 0.

A:

Q: 4. Suppose that the joint distribution of X and Y can be modeled by the following probability…

A:

Q: Let f(x, y) = 2xy 0<x< 1,0 < x + y< 1 and let it equal 0 otherwise . a. Show that f(x,y) is a joint…

A:

Q: Let X and Y have the following joint probability density function: fX,Y(x,y) = 1 / (x2 y2), for…

A: The PDF of X and Y is given as:Also, the given transformation is:U=5XYandV=8X

Q: Joint probability density function of X and Y is [1/ab 0<x<a fxy(x, y) = = {0 elsewhere and 0<y<b

A:

Q: Let Y be uniformly distributed in [0, 1], and X = In(Y+1). Find the cumulative distributionfunction…

A: For any provided value, the CDF tells the possibility that a random variable will be less than or…

Q: Suppose the joint probability density function of X and Y is given by 3 (1- Vx² + y² ) for ² + y? <…

A: From the given information,  it is clear that,  fx,y=3π1-x2+y2        for x2+y2&lt;10…

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!