4.77 A random variable X has a mean u = 10 and a variance o = 4. Using Chebyshev's theorem, find (a) P(|X – 10| 2 3); (b) P(|X – 10| < 3); (c) P(5 < X < 15); (d) the value of the constant c such that P(|X – 10| > c) < 0.04.

f (x) = 4xc for 0 ≤ x ≤ 1 ; Find the constant c so that f (x) is apdf of some random variable X, and then find the cdf, F (x) = P(X ≤ x). Sketch graphsof the pdf f (x) and the cdf F (x), and find the mean μ and variance σ2.

Q1. let Y₁ < Y₂ < Y3 < Y4 < Y5 are the order statistics of the random sample of size 5 from the distribution : f(x) = 3x², 0

Q. 4 Let X be a chi-square random variable with degree of freedom 6 (a) Find the quantile Q(0.95) and Q(0.050); (b) Let Y be a gamma random variable with a =3 and ß =- Then find 4 P(3.28< X < 25.2).

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Find the standard deviation. P(x) x*P(x) (x - )?P(x) 0.1 2(0.1) = 0.2 | (2 – 5.4)²(0.1) = 1.156 4 0.3 4(0.3) = 1.2 | (4 – 5.4)²(0.3) = 0.588 6 0.4 6(0.4) = 2.4 (6 – 5.4) (0.4) = 0.144 %3D 8 0.2 8(0.2) = 1.6 | (8 – 5.4)(0.2) = 1.352 %3D %3D Table 4.24

Suppose X and Y are independent, X has mean 5 and variance 9, Y has mean -3 and variance 6. Compute a. E[(3X + 2Y)(X + 4Y + 7)]. b. Var [6XY]

(a) Let Y be a random variable distributed as X. Determine E(Y) in terms of r. (b) Let {X1, X2, . .. , Xn} be a random sample drawn from a normal distirbution with mean u and 1 variance o?. Denote S E-(X; – X)² as the sample standard deviation. Use the 1 n - result in part (a), or otherwise, to find E(S). (c) Find an unbiased estimator for the population standard deviation o.

Let X be a random variable with pmf P(O)=0.2, P(1)-0.5, P(2)=0.3. Let F(X) be the cdf of X. (a) Find P(X<0.7) (b) Find F(1.3). (c) Find F(3).

Consider a random sample X1,..., X, with n > 3 from the exponential distribution with parameter ) > 0 and density 1 f(x; A) = x > 0. Consider the following two estimators of X: 3 1 ΣΧ, Ã = nX(1). 3 i=1