If X is a random variable, prove that Cov(X,X) = σX².

Q: Let X, Y be independent Uniform (0, 2) random variables. Define Y, if X }. Z = Compute E[Z] and…

A: X,Y ~ Uniform(0,2) E(X) = E(Y) = 0+22=1 Var(X) = Var(Y) = (2-0)212=412=13 Z = Yif X <122Yif X≥12…

Q: If X and Y are two random variables, then the covariance is Cxy = Ryy – XY

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Q: Let X ~ N(0, 2) and Y ~ covariances Cov(X,Y) and Cov(Y, X +Y). Exp(A = 3) be two uncorrelated random…

A: At first remind that                                      Independent ⇒UncorrelatedBut, Uncorrelated…

Q: Let x and y be two independent N(0, 1) random variables and Z = 1+ X + XY², W = 1+ X. Find Coy(7,W).

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Q: Let X be a continuous random variable symmetric about Y. Let Z = 1 if X>Y OR Z = 0 if X <= Y. Find…

A: Given: Z=1 if X&gt;Y OR Z=0 if X&lt;=Y

Q: Suppose X and Y are random variables with E[XY ] = 6, E[Y ] = 4 and E[X] = 5 Find Cov(X; Y )

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Q: Prove that if 'X' and ʼY' are 'random variables taking real values then [E(XY)}]SE[X°] • E[Y²].

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Q: If X and Y are two random variables, then the covariance of x+a,Y +b,where a and b are constant is…

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Q: Prove that cov(X, Y) = cov(Y, X) for both discreteand continuous random variables X and Y.

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Q: 2. Let X and Y be random variables such that V(X) = V(Y). Show that Cov(X + Y, X - Y) = 0.

A: we have given that  V(X)=V(Y) and C(X,Y)=C(Y,X)   ,C(X,X)=V(X) ,  C(Y,Y)=V(Y)

Q: Let X be a uniform random variable over the interval 1 to 4 and Y is exponential with a mean of 2.…

A: Given information: X~Uni(a=1,b=4) Standard deviation of X is: σX=b-a12       =4-112       =0.8660…

Q: Suppose that X, and X2 are aiscrete random variables with joint pdf of the form f(x1,x2) = c(x1 +…

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Q: Let X and Y be independent random variables, prove that Var (XY) = Var (X) Var (Y) if E [X] = E [ Y]…

A: Answer: For the given data, We have,

Q: Let U and V be uncorrelated random variables. Consider U = X + Y and V = X. a. Find Cov(U,V). b. Is…

A: Given that  Let U and V are uncorrected  U=X+Y and  V=X  We have to find  a..Cov(U,V)  b..is there…

Q: Find Pr (0 < X1 <1/3,0 < X2 < ½) if the random variables X1and X2have the joint p.d.f. S(X1, X2) =…

A: From the given information, fx1, x2=4X1(!-X2), 0&lt;X1&lt;1, 0&lt;X2&lt;1 Consider, P0&lt;X1&lt;1/3,…

Q: a) Let X, Y and Z be random variables. Show Cov(X + Y,Z) = Cov(X,Z) * Cov(Y,Z). %3D

A: Note: Hey there! Thank you for the question. Each of the questions (a), (b), (c), (d), and (e) is a…

Q: If Y is a chi-square random variable with v degrees of freedom, then u = E(Y) = v and a = v(Y) = 2v.…

A: Given that  If Y is a chi-square random variable with v degrees of freedom, then we have to prove…

Q: Let us say that we have jointly distributed random variables X and Y with E(X) = 3, E(Y) = 5, Var…

A:

Q: Let the joint pdf of random variables X, Y be fx,y (r, y) = axye-2-3y, for all æ > 0, y 2 0. Find a,…

A: We have given that a joint pdf of variables X and Y.

Q: X... X, are random variables, then prove that yar| X = var (X) + 2 cov (X, X). j+k k-1

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Q: Let X and Y be random variables such that var(X) = 16, var (Y) = 9, and p = cor(X, Y) = -0.5. a.…

A: Solution

Q: Let X and Y be two (continuous or discrete) random variables with E(X)=D2, E(Y)3D5, Var(X)=9,…

A: Given: E(X) = 2 E(Y) = 5 Var(X) = 9 Var(Y) = 16 Cov(X,Y) = 5 Formula Used: Var(X-Y) = V(X) - V(Y) -…

Q: Is cov(X, Y) random?

A: Given :Is cov(X, Y) random?

Q: If X and Y are any random variables with COV(X, Y) = 0.25, o = 0.36, and o = 0.49, then the…

A: From the given information, Covx,y=0.25         σ2x=0.36         σ2y=0.49 Thus,

Q: Suppose you have joint random variables X and Y. a) If a problem does not state anything about…

A: X and Y are 2 R.V. and if the dist. of X is not influenced by Y values then 2 R.V. are independent.

Q: Let X and Y be jointly Gaussian random variables with PDF exp { (12 + 4y² – 2x + 1) } fx,y(x, y) for…

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Q: 3. Let (X,Y) be a bivariate random variable, and let a and b be real constants. Show that (a)…

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Q: Let X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) +…

A: The covariance between two variables is given as follows (provided the expectation of random…

Q: Let X and Y are independent uniform random variables supported on [0, 1]. Find the proba- bility…

A: Since, X and Y are independent uniformly distributed on [0,1], then the pdf's of X and Y are fXx=1…

Q: Let X and Y be random variables. Suppose Var(X) = 1.6, Var(Y) = 1.5, and Cov(X, Y) = 0.1. Let Z =…

A: Solution: From the given information, Var(X) =1.6, Var(Y) =1.5 and Cov (X, Y) =0.1. Let Z =…

Q: Let X1, X2, ... be independent random variables with 0 mean. Let Zn Show that {Zn, n > 1} is a…

A: The objective is to show Zn =∑Xiis a martingale.

Q: Let V=X+Y and U=X-Y are random variables, then the condition which make the covariance

A: Given that V=X+Y and U=X-Y are random variables. Formula for covariance between U and V is…

Q: If Y is an exponential random variable with parameter , then μ = E(Y) = 8 and o² = V(Y) = 8². The…

A: We have given that Y is an exponential random variable with parameters beta then mean = E(Y) = β…

Q: If X, Y, Z, and W are random variables, then show tha (a) Cov(X + Y, Z) = Cov(X, Z) ± Cov(Y, Z). tud

A: Given that, If X, Y, Z, and W are random variables.

Q: Verify central limit theorem for the independent random variablesX,, where for each k, P{X =± 1} = .

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Q: Let X1, X2, X3 be random variables each having a mean u and variance o'. Further, Cov(X1 X2) 2,…

A: The mean of a constant is equal to the value of the constant and the mean of sum of random variables…

Q: Show that if X and Y are independent Exp(a)-distributed random vari- ables, then X/Y e F(2, 2).

A: The distribution of a continuous random variable can be identified by its pdf because the pdf of…

Q: Suppose (X,Y) is a continuous random variables with joint probability d 4xy Osxs1,0sys1 x,Mx.v) = {"…

A: Let X be a random continuous variable with probability density function(pdf) f(x) then the expected…

Q: Let X,, X2, .. .be independent Cauchy random variables, each with PDF d f(x)= T(d² +x²)' Show that…

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Q: Two random variables Z and W are defined as Z = X + aY and W = X - aY, where X and Y are also random…

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Q: Show that if X is not a deterministic random variable, then H(X) is strictly positive. What happens…

A: Suppose a random variable takes values on k numbers {a1 , a2,...., ak} with corresponding…

Q: Q. For any random variables X and Y and the constant a, b, c and d show that Cov(aX + b, cY + d) =…

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Q: e Prove that E(X) independent. E(X/Y), where X and Y are two random variables which are %3D

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Q: Suppose X is a discrete random variable with pmf defined as p(x) = log10 fo x = {1,2,3,...9} Prove…

A: A pmf is legitimate if (i) 0≤P(x) ≤1                                   (ii) ∑P(x) =1

Q: Let X and Y be two random variables such that Cov(X.Y) = -3 . Then %3D O None of these O…

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Q: 4. Let X, Y be independent random variables. Prove that E(XY) = E(X) E(Y) and hence show that V(X…

A: The expected value is also known as average of the probability distribution function. It is the…

Q: Let X be a random variable with p.d.f f(x) =x(9-x), 0sxS 3. Find the mode.

A: Introduction Mode: Mode is one of the types of measures of central tendency. Mode is the location…

Q: If random variables X1 and X2 are independent then .A o=(°x'x)aq »(x,x,)="," cov (X ,X_)=0 cov 2 .B…

A: Given,the random varibles X1 , X2 are independent .

Q: Theorem 11. Let X be a random variable and let g(x) be a non-negative function. Then for r > 0,…

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Q: random variables with E(X) =2, E(Y)=3 ,V(X)=4,V(Y)=16. Find a)E(5X-Y) b)V(5X-Y) c)COV(3X+Y,Y)…

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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!