2x A random variable X has the density function f(x)= } k- 0, Otherwise. For what value of k the variance of X equal to 2.

Q: Suppose Y1,..., Yn are independent random variables with probability mass function f(y) = p" (1 –…

A: Given that  Let y1 y2 y3.....yn are independent random variables with probability mass function…

Q: iven an n dimensional random sample X from a population with density f(x,6

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Q: Assume that X and Y are two independent random variables, both exponentially distributed with the…

A: X and Y are independent random variables, both exponentially distributed with parameter 1. The…

Q: Suppose X has a continuous uniform distribution over the interval -1,1. Determine the variance.…

A: GivenX has a continuos uniform distribution over the interval (-1,1)The corresponding density…

Q: The discrete random variable X has probability mass function kx x 2,3 x+1' 2kx P(X =x) = x=4,5 x-1'…

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Q: In a certain country, the amount of annual rainfall (in tonnes per m) is modelled through a Gamma…

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Q: Suppose we have test of hypothesis about the mean of a normal distribution with known variance 1. Ho…

A: Given: Hypothesized mean = 3 R=-∞,-2∪2,∞ Graph:

Q: x is a uniform random variable in the interval (x,, x,). Find the expected value of x.

A: ANSWER: For the given data,

Q: Let X be a random variable from an exponential distribution with parameter 1/2. Calculate the…

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Q: 1. If the random variable X from a distribution with mean 50 and variance 10 and y = 0.8x + 15. Find…

A: It is an important part of statistics. It is widely used.

Q: Two lives aged x and y take out a policy that will pay £15,000 immediately on the death of (x)…

A: For the benefit of 15000 to be paid, ⇒ x must die between 5 years  and 15 years  after y. Let Z…

Q: Suppose X1,  X2, X3 denote a random sample from the exponential distribution with density function.…

A: A continuous random variable X is said to follow the exponential distribution if its probability…

Q: Among all distribution laws of a continuous random variable X with the same known variation D, find…

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Q: The random variable x is known to be uniformly distributed between 10 and 20.a. Show the graph of…

A: Parameters of uniform distribution: a = 10, b = 20. Hence, probability density function of uniform…

Q: The time it takes to repair a personal computer is a random variable whose density, in hours, is…

A: Let X be a continuous random variable with probability density function fx. The expected value of…

Q: If the continuous random variable X has the probability density ( 62(1 - 2) for 0<z<1 elsewhere f(2)…

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Q: 20. A continuous random variable X has CDF given by if x <1 F(x) ={2(x – 2 + 1/x) if 1<x<2 if 2 < x…

A: As Prof. A. K. Giri defined, "Probability Density Function: A continuous function f(x) is said to be…

Q: 1. If the probability density of the random variable is given by 630x (1- x) for 0<x<1 flx) =…

A: Given: The distribution given is as follows fx=630 ×x4×1-x4             0&lt;x&lt;1

Q: If a dealer’s profit, in units of $5000, on a new automobile can be looked upon as a random variable…

A: The E(X) is calculated as follows: E(X)=2∫01x(1-x)dx=2∫01(x-x2)dx=2(x22-x33)01 =2(12-13)=13

Q: b) Derive the Fisher information I„(0) of this distribution.

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Q: If X is a continuous random variable with probability density function: (2-x) 0<x<2 f(x) = {1"…

A: Step 1: Given the probability density function,                                   f(x)= 34x(2-x)…

Q: 2. A system consisting of one original unit plus a spare can function for a random amount of time X.…

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Q: A system for a random amount of time X (in units of months) is given by a density xe 2 ; x>0,…

A:

Q: A random variable X has a Pareto distribution if andonly if its probability density is given by f(x)…

A: Note: The pdf is unclear in the question, hence solved for a two-parameter Pareto distribution: The…

Q: circle of radius r has area A = ar². If a random circle has a radius that is uniformly dis- puted on…

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Q: X and Y be standard normal random variables with correlation ρ. Compute the joint and marginal…

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Q: 3.For the continuous random variable X with probability density function defined by ¤(2 – x) 0 <x <…

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Q: et X1,..., Xn be a random sample from a uniform distribution on the interval [20,0], where 0. et…

A: Solution

Q: 4. Let X be a lifetime (in hours) of a lightbulb and suppose that X has a PDF of e0.001x f(x) = if x…

A: Solution According to the guidelines we solve only three subparts

Q: 2) The proportion of time, X, that a manufacturing machine is in operation during an 8-hour shift is…

A: Solution

Q: 2 If the probability density of a random variable is given by (k + 2)x³ 0 < x < 1 f(x) = 0.…

A: f(x)=(k+2)x3        0&lt;x&lt;10                           o/wproperities∫f(x)dx=1

Q: Let X1,..., Xn be iid random variables with the density function f(x;0) = (0+ 1)xº for 0 <x < 1. (a)…

A: Let X1,X2,...Xn be iid random variables with pdf given by, f(x,θ)=(θ+1)xθ      ; for 0≤x≤1

Q: The X be a continuous random variable with cumulative distribution function 1 F(x) = 1 for x > 0…

A: From the given information, X be continuous random variable with cumulative distribution is,…

Q: A system for a random amount of time X (in units of months) is given by a density Stxe¯² ; x>0,…

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Q: 2.1. Let X1, X2. .... Xm denote a random sample from the exponential density with mean , and let Y,,…

A: Given X1, X2,..... Xm And Y1, Y2,.... Yn Two samples Lilehood ratio criterion =?

Q: An instrument is used to measure very small concentrations, X, of a certain chemical in soil…

A: From the information, observe that an instrument is used to measure very small concentrations of…

Q: For a statistic to be a good estimator of a parameter, two properties it must satisfy are…

A: It is given that Xi~Expθ; i=1,2,3 with  EXi=θ, EX2i=2θ2, and VarXi=θ2. The sample mean of  X is…

Q: 6) Allow a continuous probability distribution to be defined as f(x) = 4x³ for the range 0 SXS1.…

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Q: 11. Suppose random variable X is distributed as normal with mean 2 and standard deviation 3 and…

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Q: A system for a random amount of time X (in units of months) is given by a density (txe2 : x>0, :…

A: The moment generating function is defined as MX(t)=EetX

Q: If X = ln Y has a normal distribution with the mean μand the standard deviation σ, find the…

A: Normal Distribution pdf is given by ,  It is given that ,  X = ln Y  that is ,  Consider , Using…

Q: Suppose we obtain n observations from a sinusoid contaminated by Gaussian noise: X = A sin(2 · r fk+…

A: The obtained n observations from a sinusoid contaminated by Gaussian noise are: Xk=A·sin2πfk+ϕ+Wk,…

Q: f X is a random variable that is uniformly distributed between -1 to 1, find the PDF √|X| of and pdf…

A: Solution:

Q: 2. Determine if a Normal (u, o2) belongs to a 2-parameter exponential family of densities. Derive…

A: Determine if N(mu,sigma2) belongs to 2 parameter exponential family of distribution . Drive…

Q: |The amount of electricity consumed (in kWh) on a randomly chosen day in a warehouse can be modelled…

A: Let , the random variable X be the amount of electricity consumed on a randomly chosen day in…

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!