ution whose relative frequency density is given y: ind mean, variance, B1, and B2 and hence show that the distribution is symmetrical. Also (i) End mean deviation about mean, and (ii) show that for this distribution uz+1 = 0, and (iii) End the mode, harmonic mean and median. f(x) = yo- x (2- x), 0SxS 2, %3D
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- 13) Random variables X and Y have joint pdf fXY={4xy, 0≤x≤1, 0≤y≤1fXY={4xy, 0≤x≤1, 0≤y≤1 Find Correlation and CovarianceA random variable X is normally distributed with a mean of 100 and a variance of 100, and a random vari- able Y is normally distributed with a mean of 200 and a variance of 400. The random variables have a corre- lation coefficient equal to 0.5. Find the mean and vari- ance of the random variable: W = 5X + 4Y1. Define a random variable X as the proportion of time that a student spends on a certain test. Research suggests that the pdf of x is f(x) = (θ + 1)xθ , 0 ≤ x ≤ 1. For the parameter θ, we know θ > −1, but its exact value is still unknown. To estimate the parameter, a random sample of 10 students is obtained, and their proportions of time are: 0.92, 0.79, 0.90, 0.65, 0.86, 0.47, 0.73, 0.97, 0.94, 0.77. Find the Maximum Likelihood Estimator (MLE) of θ.
- Which of the following is a characteristic of latent variables? A. The value of the latent variable is high, than the dependent variable for that observation is likely to be 1. B. They are normally distributed. C. The value of the latent variable is high, than the dependent variable for that observation is likely to be 0. D. It is an observed continuous variable reflecting the propensity of an individual observation of Yi to be equal to 0 or 1.A person is interested in constructing a portfolio. Two stocks are being considered. Letx = percent return for an investment in stock 1, and y = percent return for an investment instock 2. The expected return and variance for stock 1 are e(x) = 8.45% and Var(x) = 25.The expected return and variance for stock 2 are e(y) = 3.20% and Var(y) = 1. Thecovariance between the returns is sxy = −3.a. what is the standard deviation for an investment in stock 1 and for an investment instock 2? Using the standard deviation as a measure of risk, which of these stocks isthe riskier investment?b. what is the expected return and standard deviation, in dollars, for a person who invests$500 in stock 1?c. what is the expected percent return and standard deviation for a person who constructsa portfolio by investing 50% in each stock?d. what is the expected percent return and standard deviation for a person who constructsa portfolio by investing 70% in stock 1 and 30% in stock 2?A person is interested in constructing a portfolio. Two stocks are being considered. Letx = percent return for an investment in stock 1, and y = percent return for an investment instock 2. The expected return and variance for stock 1 are e(x) = 8.45% and Var(x) = 25.The expected return and variance for stock 2 are e(y) = 3.20% and Var(y) = 1. Thecovariance between the returns is sxy = −3.a. what is the standard deviation for an investment in stock 1 and for an investment instock 2? Using the standard deviation as a measure of risk, which of these stocks isthe riskier investment?b. what is the expected return and standard deviation, in dollars, for a person who invests$500 in stock 1?c. what is the expected percent return and standard deviation for a person who constructsa portfolio by investing 50% in each stock?d. what is the expected percent return and standard deviation for a person who constructsa portfolio by investing 70% in stock 1 and 30% in stock 2?e. Compute the correlation…
- 9.1) Suppose X1, X2, and X3, denotes a random sample from the exponential distribution with density function shown in the image. a) Which of the above estimators are unbiased for θ? b) Among the unbiased estimators of θ, which has the smallest variance?1)Let x be a random variable Gaussian with zero mean and variance 1. Find:a)The conditional pdf and pdf of x given x > 0;b)E [ x| x>0 ]c)Var [ x | x >0]Let X ,X , ,X 1 2 n … be a random sample of size nfrom a population with mean μ and variance σ2.(a) Show that X2 is a biased estimator for μ2.(b) Find the amount of bias in this estimator.(c) What happens to the bias as the sample size n increases?
- 2) Let G and H be two independent unbiased estimators of θ. Assume that the variance of G is two times the variance of H. Find the constants a and b so that aG + bH is an unbiased estimator with the smallest possible variance for such a linear combination.If the probability density of X is given by f(x) =2x−3 for x > 10 elsewherecheck whether its mean and its variance exist.3.7. Consider the performance function Y- 3X1-2X2 where X1 and X2 are both normally distributed random variables with Rx.-16.6 Ох.-2.45 μΧ.-18.8 ơXs: 2.83 The two variables are correlated, and the covariance is equal to 2.0. Determine the probability of failure if failure is defined as the state when Y0.