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1. State three properties of the normal distribution. [3 marks) (a) Suppose that X is a...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y)...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the inte...
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the interval I-1, 1]. Recall that this means the density function fu satisfies for(z-a: a.crwise. 1 u(z), -1ss1, a) Find thc cxpccted valuc and the variancc of U. We now consider estimators for the expected value of U which use a sample of size 2 Let Xi and X2 be independent random variables with the same distribution as U. Let X = (X1 +...
Please show steps (and formulas) for part b Problem 2. a. X has a normal distribution...
Please show steps (and formulas) for part b
Problem 2. a. X has a normal distribution with mean 5 and variance 25. Y has a normal distribution with mean 3 and variance 16. In addition, X and Y are independent. If W = X+Y, find P(W > 9). b. Random variables U, V, Z are such that E[U] = 1, E[V] =5, E[2] = -3, Var[U] = 1, Var[V] = 4, Var[2] =1, Cov[U,V] =-1,Cov[U, 2] = 2, Cor[V, 2]...
Let X be a random variable with the following probability distribution: Value x of X P(X=x)...
Let X be a random variable with the following probability distribution: Value x of X P(X=x) 0.15 0.10 3 0.05 0.05 0.30 0.35 Find the expectation E (X) and variance Var (X) of X. (If necessary, consult a list of formulas.) E (x) = 0 x 6 ? var(x) -
Let X be a random variable with the following probability distribution: value x of X P...
Let X be a random variable with the following probability distribution: value x of X P (X= x) 40 50 60 70 80 90 0.10 0.15 0.40 0.20 0.05 0.10 Find the expectation E (X) and variance Var(X) of X. (If necessary, consult a list of formulas.) Var(x)-
Fill in the P(X = x) values to give a legitimate probability distribution for the discrete...
Fill in the P(X = x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are 2, 3, 4, 5, and 6. Value I of x P(x = x) 2 0.16 3 4 0.17 0.29 6 0 X 6 For Subm Let X be a random variable with the following probability distribution: 1 Value x of X P(X=x) 0.25 2 0.05 3 0.15 4 0.15 5 0.10 6 0.30 Find the expectation E...
3. Suppose X,,X2,, is a random sample from a standard normal distribution and let Z be...
3. Suppose X,,X2,, is a random sample from a standard normal distribution and let Z be another standard normal variable that is independent of X,X, X, UX , and V X- 9 9 Let X (x - x)2 i-1 Determine the distribution of each of the variables X, U and V. (a) (b) Determine the distribution of the variable 32 VU Determine the distribution of the variable (c) (d) (e) Determine the distribution of the variable y (where Y is...
4.) a.) Suppose that X is a normal random variable with mean 4. If P[X >...
4.) a.) Suppose that X is a normal random variable with mean 4. If P[X > 9} = 0.1 approximately what is Var(X)? (15 points) b.) Measure the number of kilometers traveled by a given car before its transmission ceases to function. Suppose that this distribution is governed by the exponential distribution with mean 800,000. What is the probability that a car's transmission will fail during its first 40,000 kilometers of operation? (10 points)
Let X be a random variable with the following probability distribution: Value x of X P(...
Let X be a random variable with the following probability distribution: Value x of X P( xx) 0.40 5 0.05 6 0.10 0.35 В 4 7 0.10 Find the expectation E(X) and variance Var (x) of X. (If necessary, consult a list of formulas.) х 5 2
3) Suppose X,,X,,X, (n > 1) is a random sample from Bernoulli distribution with Circle out...
3) Suppose X,,X,,X, (n > 1) is a random sample from Bernoulli distribution with Circle out your Class: Mon&Wed or Mon.Evening p.mf. p(x)=p"(I-p)'-x , x = 0,1, , thenyi follows ( ). Binomial distribution B(a.p) eNormal distribution N(p,mp(- O Poisson distribution P(np) Dcan not be determined. 4) Suppose X-N(0,1) and Y~N(24), they are independent, then )is incorrect. X+Y N(2, 5) C X-Y-NC-2,5) BP(Y <2)>0.5 D Var(X) < Var(Y) x,X,, ,X, (n>1) is a random sample from N(μσ2), let-1ΣΧί 5) Suppose...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the interval I-1, 1]. Recall that this means the density function fu satisfies for(z-a: a.crwise. 1 u(z), -1ss1, a) Find thc cxpccted valuc and the variancc of U. We now consider estimators for the expected value of U which use a sample of size 2 Let Xi and X2 be independent random variables with the same distribution as U. Let X = (X1 +...
Please show steps (and formulas) for part b
Problem 2. a. X has a normal distribution with mean 5 and variance 25. Y has a normal distribution with mean 3 and variance 16. In addition, X and Y are independent. If W = X+Y, find P(W > 9). b. Random variables U, V, Z are such that E[U] = 1, E[V] =5, E[2] = -3, Var[U] = 1, Var[V] = 4, Var[2] =1, Cov[U,V] =-1,Cov[U, 2] = 2, Cor[V, 2]...
Let X be a random variable with the following probability distribution: Value x of X P(X=x) 0.15 0.10 3 0.05 0.05 0.30 0.35 Find the expectation E (X) and variance Var (X) of X. (If necessary, consult a list of formulas.) E (x) = 0 x 6 ? var(x) -
Let X be a random variable with the following probability distribution: value x of X P (X= x) 40 50 60 70 80 90 0.10 0.15 0.40 0.20 0.05 0.10 Find the expectation E (X) and variance Var(X) of X. (If necessary, consult a list of formulas.) Var(x)-
Fill in the P(X = x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are 2, 3, 4, 5, and 6. Value I of x P(x = x) 2 0.16 3 4 0.17 0.29 6 0 X 6 For Subm Let X be a random variable with the following probability distribution: 1 Value x of X P(X=x) 0.25 2 0.05 3 0.15 4 0.15 5 0.10 6 0.30 Find the expectation E...
3. Suppose X,,X2,, is a random sample from a standard normal distribution and let Z be another standard normal variable that is independent of X,X, X, UX , and V X- 9 9 Let X (x - x)2 i-1 Determine the distribution of each of the variables X, U and V. (a) (b) Determine the distribution of the variable 32 VU Determine the distribution of the variable (c) (d) (e) Determine the distribution of the variable y (where Y is...
4.) a.) Suppose that X is a normal random variable with mean 4. If P[X > 9} = 0.1 approximately what is Var(X)? (15 points) b.) Measure the number of kilometers traveled by a given car before its transmission ceases to function. Suppose that this distribution is governed by the exponential distribution with mean 800,000. What is the probability that a car's transmission will fail during its first 40,000 kilometers of operation? (10 points)
Let X be a random variable with the following probability distribution: Value x of X P( xx) 0.40 5 0.05 6 0.10 0.35 В 4 7 0.10 Find the expectation E(X) and variance Var (x) of X. (If necessary, consult a list of formulas.) х 5 2
3) Suppose X,,X,,X, (n > 1) is a random sample from Bernoulli distribution with Circle out your Class: Mon&Wed or Mon.Evening p.mf. p(x)=p"(I-p)'-x , x = 0,1, , thenyi follows ( ). Binomial distribution B(a.p) eNormal distribution N(p,mp(- O Poisson distribution P(np) Dcan not be determined. 4) Suppose X-N(0,1) and Y~N(24), they are independent, then )is incorrect. X+Y N(2, 5) C X-Y-NC-2,5) BP(Y <2)>0.5 D Var(X) < Var(Y) x,X,, ,X, (n>1) is a random sample from N(μσ2), let-1ΣΧί 5) Suppose...
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