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- EX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.4. If XN3 (μ, 2) where uT=[2-3 1] and E= 1 3 2 1 2 21 then find the distribution of 3X₁ - 2X2 + X3, where a= [3 -2 1].Q 4.1. Suppose X and Y are independent random variables, which are jointly continuous. 1. Show carefully that the distribution of ‘X + Y given X = x' is equal to the distribution of the random variable x + Y. Hint: Consider the change of variables h(x, y) = (x,x+y).
- 2. Suppose that X is normally distributed with u = -7 and o? P(X 2x) = 0.84. = 14. Find the value of x for which4. 65th Pg. 59, 1. 7. 19. * Find the of the distribution having pot F(x) = 1x1/4₂ Percentile where -a, < x < 2, f(x) = 0, elsewhere.4.1. Suppose X and Y are independent random variables, which are jointly continuous. x' is equal to the distribution 1. Show carefully that the distribution of 'X + Y given X of the random variable à +Y. Hint: Consider the change of variables h(x, y) = (x,x+y). = 2. Suppose X and Y each have an exponential distribution with parameter a, find the joint p.d.f of X and X + Y. 3. In the situation of (ii) above calculate E[X | X + Y].
- 4. Suppose that X has pdf f(x) = 3x² for 0 < x< 1. Find the pdf of the random variable Y = VX.If X1, X2, ... , Xk have the multinomial distribution of Definition 8, show that the mean of the marginal distribu-tion of Xi is nθi for i = 1, 2, ... , k.8. Given the Beta Distribution where p(x) = B(a,b).ªªa- Calculate the mean and variance B(a, b) = -xª-¹ (1-x)b-1 1 = √₁² 2²- x-¹(1-x)-¹dx = r(a) I (b) I(a + b)