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

Q: Cuppose that X = (X1, X2) ~ N2(µ, E) and let f(x1,x2) be the joint density of X1 and X2. Prove that…

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Q: 5)? a What is P(Y₁ b What is P(Y₁+ Y₂ < 3)?

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Q: a. What value of c will make f(x) a valid density? b. What is P(X = 1)? What about P(X = 1 or X =…

A: The value of c that makes probability density function valid is obtained below: From the given…

Q: Suppose f(x)=3x2, 0<x<1 is a density function of X.  Find P(1/2<X<2).

A: We have given that,  Let x is a random variable having probability density function is,  f(x) =…

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: )Suppose X is continuously uniformly distributed on [1, 4]. Let Y = ln(X). What is the density…

A: X~Uniform[1,4]

Q: The value of c

A: The given density function is, fx=cx2      if 0&lt;x&lt;30          otherwise It is known that the…

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.

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Q: Assuming that X and Y have joint distribution f (x, y) = cx for -x < y < x, 0 < x < 1, and f(x, y) =…

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Q: (a) Show that f(x) = 3x2 for 0 &lt; x &lt; 1 represents a density function.(b) Sketch a graph of…

A: Any function is a probability density function if it satisfy following two condition  Now, lets…

Q: Suppose that X ~ · Uniform(a, b) with 0 < a < b < o. Find the probability density function of Y,…

A: Given information: X~Uniforma,b fx=1b-a, 0&lt;&lt;b&lt;∞ Given that Y=X Consider,…

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: . Let f (x) = cx 3 and S X = [0, 2] (X is continuous and c is a constant). a. What value of c will…

A: The value of c that makes probability density function valid is obtained below: From the given…

Q: Let random variable X have the density function: cx² + x, 0<x<1, f(x)%3= 0, otherwise. Find the…

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Q: Let Y, and Y, have the joint probability density function given by f(y1, Y2) = - {k(1 - y2), 0 sy1…

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

A: From the given information, fx,y=651+x+y, 0≤x,y≤1, x+y≤1 g) Consider, fx=∫yfx,ydy…

Q: Let Z ∼ N(0,1) and Y = Z1/3 (a) Find the density function (pdf) of Y . b) FindP (Y> (1/4)^1/3))

A: Given:- Let Z ∼ N(0,1)  Y = Z^1/3

Q: *9. Consider the T, distribution: t2\ - 1+ VUTT (5) レ+ f(t;v) = Let y(x) denote the standard…

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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: If X and Y are jointly continuous with joint density function fx,y(z,y), show that X + Y is…

A: Let X and Y are jointly continuous with joint density function fX,Y(x,y) We have to prove that X+Y…

Q: Assuming that X and Y have joint distribution ƒ (x,y) = cx for –x < y < x, 0 < x < 1, and f(x, y) =…

A: Suppose that X and Y have joint probability distribution f(x,y)=cx for -x&lt;y&lt;x,0&lt;x&lt;1…

Q: A continuous random variable X that can assume values between x = 2 and x=6 has a density function…

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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 and Y be continuous random variables with joint density function { if x E R, y > 0 f(x, y) =…

A: # Joint pdf of random variable x and y is given f(x,y)=e^-(y+x^2)/sqrt(π) : x from real number and…

Q: 1. Consider the density function f(x) = {k√x₁ 0, 0 < x < 1, elsewhere. a) Evaluate k. b) Find F(x)…

A: The random variable X has probability density function is,The question is required to calculate the…

Q: 1 /x_ar. x > 0 (a) f(x) = otherwise |c(1 – x²) -1 5 (c) f(x) = { x² x < 5

A: E(X)= integration|0infinite x*f(x)dx = Integration|0infinite x2e-x/2/4 dx Using gamma integral…

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: The density function of X is:  f(x) = c·x2 ·( 1 - x ).  Calculate the mode of X.

A: The density function of X is: f(x) = c·x2 ·( 1 - x ).  Calculate the mode of X.

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

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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!