Suppose fz(z) is a function given by So23 – 2) if -<<0 fz(2) = otherwise for a fixed constant c. Can fz(z) be a density function? Explain.

Q: Differentiate f(w) = sinw + w²tan-1w

A: We have to find differentiation of given function.

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: Suppose the volume of a tree is given by V=1c²h where c is the circumference of the tree in meters…

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Q: Find the distribution function, G(U) transformation, and show that P(U<=u) is the same as the…

A: The provided information is as follows:The probability density function is calculated as follows:

Q: Let X and Y has joint probability density function f(X, Y) = 81 - , 0 <x< 3, 0 <y< 3. Find E(X).

A: The joint probability density function is, fX,Y=x2y281,0&lt;x&lt;3,0&lt;y&lt;3

Q: The random varible X has density function f(x)= cxk+1(1-x)k for 00 and 1<k<2. What is mode of X?

A: This pdf is the beta first kind  We want to find the mode of beta first kind

Q: Find the mass of a thin wire with mass density function 8(x, y, z) = x2 + xy + 2² if the wire is in…

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Q: Q1 Let (X₁, X₂) be jointly continuous with joint probability density function " f(x1, x₂) = x1 0 <…

A: Let  be jointly continuous with joint probability density function:

Q: Is the function defined as follows a density function? fx (x) = 0 for x 4

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Q: If f(x) is density function then f(x)<0.

A: Given f(x) , a density function we need to find if given statement is true / false

Q: Is the function defined as follows a density function? fx (x) = 0 for x < 2 (3 + 2x) 18 for 2 <x< 4…

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Q: Let X1, , X, be iid with population pdf given by h ( which integrates to 1 over the real line. Here,…

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Q: Find a constant b > 0 so that the function e3x Sx (x) = { 4 0 < X<b elsewhere is valid density…

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Q: a red coin as a 1 on one side and a 3 on the other. a blue coin has a 0 and 1. let X represent the…

A: Given Red coin→ 1,3 Blue coin→ 0,1 X→ Total of both coins Y→ No. flipped on the blue coin

Q: The Rayleigh density function is sometimes used by engineers to model lengths of life of electronic…

A: a) To find the expected value (mean) of a random variable with a given probability density function…

Q: The density function of X is given by f(x) = a + bæ, 0< x < 1. (1) If E(X) = , find a and b. (2)…

A: density function of X is given by;f(x)=1+bx,  0≤x≤1

Q: One recognizes the integrand of Equation (23) as a r(a, 6) density, hence M (t) = (;)"(1) = (24) |3|…

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Q: sin 8 cos (cos a cosec 0) do 1/2-a (1-cos a).

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Q: Let I,"Erpi). f(x) = pe"1>0 IIN, -4.Find the pdf of (a) r(m.B) (b)[(1, B) (e )r(m, nB) (d) exp(nB)…

A: X~ exp(β) We want find the pdf of  Y=∑X ~ ?

Q: Let f(x, y, z) = x2 (e)^y/z. Compute f(-1, 0, 1), f(1, 3, 3), andf(5, -2, 2).

A: Let fx,y,z=x2eyz 1. we have to compute f(-1,0,1) as, fx,y,z=x2eyzHere x=-1, y=0,z=1f-1,0,1=-12e01=1…

Q: What is the joint probability density function f₁,2 of Y₁ = X₁/X₂ and Y₂ = X₂?

A: We will use Jacobian method of transformation to find the joint probability of Y1 and Y2.

Q: Suppose that X, Y are jointly continuous with joint probability density function ) = ce = ² (²+2)² 2…

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Q: Let f(x) is density function then

A: Continuous distribution: Properties for probability density function: Let f(x) be the probability…

Q: Either f,g,h € F(R, R) the functions f(x) = sin(x)^2 And g(x) = cos(x) and h(x) = sin(x). Show that…

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Q: 4. Let F(x) = S f(2)dz where f(x) is a valid probability density function. Find an expression for…

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Q: 5) For the following density function f(x) =c x2 (1 –x), 0<x<l, 1) Find the constant c 11) Find Mean

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Q: Now, assume that Y₁,..., Yn distributed iid, each with density e-(y-0), if y> 0; 0, otherwise, fy…

A: Let us assume that Y1, Y2, . . . . , Yn distributed iid, each with density function is  fYy=e-y-θ  ,…

Q: Let f(x, y) = e*Yx4 Evaluate f(3, 5)

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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!