(6) Suppose X1 and X2 are independent standard normal random variables. Define random variables Y, = X1 Y2 X1 + X2. %3D Identify the joint probability density function of Y1 and Y2.

Q: The manager of a bakery knows that the number of chocolate cakes he can sell on any given day is a…

A: Given, The probability mass function is: 0 1 2 3 4 5 1/12 1/12 3/12 4/12…

Q: X is a discrete uniform (1,3) random variable. When X=x, Y is discrete uniform (1,x). (a) Find…

A: We are given here that:P(X = 1) = P(X = 2) = P(X = 3) = 1/3 as X is a discrete uniform distribution…

Q: If the random variable X and Y have the joint probability distribution f(x, y) = x+y, 0<x< 1,0 <y<1…

A:

Q: Let the random variable Y = a + bX. Then, µy = bux. O True O False

A: Given statement isY=a+bX

Q: Let X be a Poisson random variable such that 2 P(Xe 4P(X 1) +5P(X = 2) %3D 2020/03/om (a) Find its…

A: Given that    We have to find  a..We have to find the mean and variance of X  b .. Probability that…

Q: Let X and Y be independent random variables distributed as N(-1, 1) and N(1, 1), respectively. Find…

A: X follows a  normal distribution with , µ= -1 , σ= 1. Y follows a  normal distribution with , µ= 1 ,…

Q: Random variable X has the following probability mass function: P(X=x) = (x+1)/12 for x = 0, 1, 2, 5.…

A:

Q: deviation 3 and a random variable Y is normally distributed with mean 0 and standard deviation 4.…

A: We know that the linear combination of any normal random variable is also a random variable. So, in…

Q: Let X and Y be independent standard normal random variables. Determine the pdf of W = X² + y² using…

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Q: (i) the maximum value of p.d.f. (ii) expectation of X, E(X) (iii) variance of X.

A:

Q: A random variable X takes values 0,1,2 where E(x) =1.5, var(x)=0.25 then E (3x-x²+4) =

A: Given that E(X) = 1.5, and var(X) = 0.25 The var(X) is defined as Var(X) = E(X2) - (E(X))2

Q: we are evaluating two random variables X and Y with probabilitydistribution equal to: p(x, y) = P (X…

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Q: f X is a uniformly distributed random varibale with a=9 and b=16, then Calculate the…

A: Answer: From the given data, X follows uniform distribution, a = 9   and   b = 16

Q: Let X be a discrete random variable. (i) Prove that V ar(X) = E(X2) ? (EX)2. Hint: Use the…

A: We want to prove both results

Q: Let x be a continuous random variable with PDF 3 x > 1 x4 fx (x) otherwise Find the mean and…

A: Given PDF is fX(x) = 3x4,x ≥ 10,otherwise

Q: Let x be a discrete Probability distribution Fi Cand x= -L.0,l,2+3 į Find the value of K i. Find the…

A: Discrete Random Variable  Two numbers are often used to summarize a probability distribution for a…

Q: Let x be a dis crete random variable with probability distribution fCx) =k (x² + 20) and x =…

A: i. From the given information, the probability distribution function is f(x) = K (X2 + 20) and x=…

Q: Z is the standard normal random variable E(Z) = 1 Var (Z) = 0.a O E(Z) = 1 Var (Z) = 1.b O E(Z) = 0…

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Q: Three zero mean, unit variance random variables X, Y, and 2 are added to form a new random variable,…

A: Random variables and  have zero mean and unit variances.That is:  and These variables are added to a…

Q: Find the joint probability function of (X,Y) where

A: Provided information is, two cards are drawn at random from a box which has 5 cards numbered 1, 2,…

Q: 7- Let X be a random variable with probability density function x=1,2,5 f(x)=8 O.w. (a) Find the…

A: Result used  Y=ax+b  E(Y)=aE(X)+b  V(Y)=a2V(X)  Var(X)=E(X2)-(E(X))2

Q: Let xyz be the random quantities of three standard normal distributions, corr(X,Y) = 0.3, corr(X,Z)…

A: It is given that XYZ are the three quantities of the three standard normal distributions. The…

Q: Let X denote a random variable that has a Poisson distribution with mean 1= 2. Find (a) P(X = 4) %3D…

A: Here X~ Poisson with parameter 2. We need to find the required probabilities.

Q: Suppose X₁ and X₂ have the following joint distribution X₁ 1 d 2 1/3 1/12 1/6 0 2 1/36 0 0 0 X₂ 1 0…

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Q: Z is the standard normal random variable E(Z) = 1 Var (Z) = 0 .a O %3D E(Z) = 1 Var (Z) = 1.b O

A: In the standardized form, the normal distribution has a mean equal to zero and standard deviation…

Q: Suppose the random variable X follows an Inverse Gaussian distribution with parameters u and 0. Use…

A: From the given information, the random variable X follows an Inverse Gaussian distribution with…

Q: Let X be a random variable with E[X]=2, Var(X)= 4. Compute the expectation and variance of 3– 2X .

A: Solution-: Given: E(X)=2, Var(X)=4 Our aim to find, (a) E(3-2X)=? and  (b) Var(3-2X)=?

Q: A noisy resistor presents a voltage X distributed as a Gaussian random variable with zero mean and…

A: X : voltage of a noisy resistor    Therefore,  X ~ N(mean : μ = 0, standard deviation : σ = 2 =…

Q: The pH water samples from a specific lake is a random variable X with probability density function.…

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Q: the maximum likelihood estimator of a random variable x with uniform distribution U(0, 0), 0 =…

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Q: Let Xand Y be two continuous random variables with joint probability density 3x function given by:…

A: The given value are EX=34, EX2=35, EY=38, EY2=15 and EXY=310.

Q: For a discrete random variable X if : Points ) P ( X = x ) = 0.1x , x = 2 , 3 , 5 , then E ( X2 ) =

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Q: Let x1  and x1 be independent standard normal random variables. If Y1= x1 + 2x2 and  Y2 = 2x1 - x2,…

A: We know, any linear combination of normal distribution is a normal distribution. Here, x1 and x2 are…

Q: QUESTION: Let X have the density f(r) = 2r if 0 < a <1 and f(x) = 0 otherwise. Show that X has the…

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Q: The expected value of the transformation is 0 and the variance is 1. So if Z=0.2, when u = 3 and o =…

A: The value of Z is 0.2, mean is 3, and standard deviation is 0.2.

Q: The probability that data are entered incorrectly into a field in a database is 0.008. A data entry…

A:

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!