Suppose Y = X^2 + Z^2,where X , Z are both normally distributed, but they are...
Suppose Y = X^2 + Z^2,where X , Z are both normally distributed, but they are not independent. What is the distribution of Y?
| Exponential Distribution |
| Normal distribution |
| Chi Square distribution |
| None of the other answers |
Homework Answers
If standard normal variates are independent and their sum of squares leads to chi square distribution
But here X and Z are both normally distributed where they are not independent.
Y = X^2 + Z^2
Distribution of Y is Normal distribution.
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Suppose Y = X^2 + Z^2,where X , Z are both normally distributed,
but they are...
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10.3.8 Suppose that Y = E(Y | X) + Z, where X, Y and Z are random variables. (a) Show that E (Z | X) = 0. (b) Show that Cov(E(Y | X), Ζ) = 0. (Hint. Write Z-Y-E(YİX) and use Theo- rems 3.5.2 and 3.5.4.) (c) Suppose that Z is independent of X. Show that this implies that the conditional distribution of Y given X depends on X only through its conditional mean. (Hint: Evaluate the conditional distribution function...
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A) Suppose X is normally distributed with mean 10 and standard deviation 2. Then P(X > 15) = P(Z > __) where Z is standard normal. (Fill in the blank.)
Let X and Y be two independent standard nor- mally distributed random variables, i.e., both X...
Let X and Y be two independent standard nor- mally distributed random variables, i.e., both X and Y follows standard normal function (each has mean zero and variance one). we define the random variable Z := X^2 + Y ^2. Compute Z’s density function for all real values (should be exponential with some parameter).
1.1 [Probability and Statistics] Let X and Y be jointly distributed normal random variables, where cov[X,...
1.1 [Probability and Statistics] Let X and Y be jointly distributed normal random variables, where cov[X, Y]-2 In other words, the joint distribution of the pair (X, Y) ~N(,),where 1 |.and Σ := |.-2 9 What is the distribution of the random variable Z:-X -2Y?
10. (5pt) Suppose that X and Y are two normally distributed random variables. X has mean...
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Suppose three random variables X, Y, Z have a joint distribution PX,Y,Z(x,y,z)=PX(x)PZ∣X(z∣x)PY∣Z(y∣z). Then X and Y are independent given Z? True or False Suppose random variables X and Y are independent given Z , then the joint distribution must be of the form PX,Y,Z(x,y,z)=h(x,z)g(y,z), where h,g are some functions? True or false
1. Use a normal-scores plot to see if the data are approximately normally distributed. 12, 14,...
1. Use a normal-scores plot to see if the data are approximately normally distributed. 12, 14, 18, 22, 25, 29, 31 The data (does, does not) appear to be approximately normally distributed because the normal-scores plot (is, is not) roughly linear. 2. Random variable X and Y have a bivariate normal distribution. The conditional density of X given Y = y is a a. bivariate normal distribution b. chi-square distribution c. linear distribution d) normal distribution e. not necessarily any...
Having troubles with question 2. Please help 2. If X has a Gamma distribution with parameters...
Having troubles with question 2. Please help
2. If X has a Gamma distribution with parameters a and B, then its mgf is given by (a) Obtain expressions for the moment-genérating functions of an exponential random variable and of a chi-square random variable by recognizing that these are special cases of a Gamma distribution and using the mgf given above. (b) Suppose that X1 is a Gamma variable with parameters α1 and β, X2 is a Gamma variable with parameters...
3. (20 marks) Suppose Y...Y is a random sample of independent and identically distributed Gamma(c...
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10) Suppose that X follows a chi-square distribution with m degrees of freedom and S=X+Y. Given...
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10.3.8 Suppose that Y = E(Y | X) + Z, where X, Y and Z are random variables. (a) Show that E (Z | X) = 0. (b) Show that Cov(E(Y | X), Ζ) = 0. (Hint. Write Z-Y-E(YİX) and use Theo- rems 3.5.2 and 3.5.4.) (c) Suppose that Z is independent of X. Show that this implies that the conditional distribution of Y given X depends on X only through its conditional mean. (Hint: Evaluate the conditional distribution function...
1.1 [Probability and Statistics] Let X and Y be jointly distributed normal random variables, where cov[X, Y]-2 In other words, the joint distribution of the pair (X, Y) ~N(,),where 1 |.and Σ := |.-2 9 What is the distribution of the random variable Z:-X -2Y?
10. (5pt) Suppose that X and Y are two normally distributed random variables. X has mean 2 and standard deviation !5 Y has mean 5 and standard deviation 3. Their correlation is 0.6. What is the mean and standard deviation of X + Y? What is the distribution of X+ Y? What if X and Y are jointly normally distributed? What if they are not jointly normally distributed? Explain your answer.
1. Use a normal-scores plot to see if the data are approximately normally distributed. 12, 14, 18, 22, 25, 29, 31 The data (does, does not) appear to be approximately normally distributed because the normal-scores plot (is, is not) roughly linear. 2. Random variable X and Y have a bivariate normal distribution. The conditional density of X given Y = y is a a. bivariate normal distribution b. chi-square distribution c. linear distribution d) normal distribution e. not necessarily any...
Having troubles with question 2. Please help
2. If X has a Gamma distribution with parameters a and B, then its mgf is given by (a) Obtain expressions for the moment-genérating functions of an exponential random variable and of a chi-square random variable by recognizing that these are special cases of a Gamma distribution and using the mgf given above. (b) Suppose that X1 is a Gamma variable with parameters α1 and β, X2 is a Gamma variable with parameters...
3. (20 marks) Suppose Y...Y is a random sample of independent and identically distributed Gamma(c, B) random variables. Suppose c is a known constant. a) (5 marks) Find an exact (I-a)100% CI. forty: cß based on Y. Hint: Make use of the Chi-Square distribution when finding your pivot. b) (5 marks) Find an approximate (1-α)100% CI. forMy-cß based on only Y using a n (Y-CB) ~ Normal (0, Normal approximation and the pivot Z- c) (5 marks) Find an approximate...
10) Suppose that X follows a chi-square distribution with m degrees of freedom and S=X+Y. Given that S follows a chi-square distribution with m+n degrees of freedom, and X and Y are independent, show that y follows a chi-square distribution with n degrees of freedom.
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