Problem 3.2: Let the p.d.f. of the continuous random variable X be the expression below. Compute: (c) E3-2X] (e) σχ (a) c (b) E[X] (d) Var(X) Jesin(Tr) for 0 < r ≤0.694 f(x) = elsewhere
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![Problem 3.2: Let the p.d.f. of the continuous random variable X be the
expression below. Compute:
(e) σχ
(a) e
(b) E[X]
(c) E3 - 2X]
(d) Var(X)
[csin(rr) for 0 < r ≤0.694
2) - {cxin(x²)
elsewhere
f(x) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6358c0a1-447c-4d6f-94b0-16166908ab3e%2Fa4c45ae2-34ad-4d6f-895c-f4ff47ebbf89%2F05mgcxf_processed.jpeg&w=3840&q=75)
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- 4. Suppose X₁ and X₂ are independent random variables with cdf Fx(x) = sin(x), 0 ≤ x ≤ 1/ a. Show that fx(2) (x) = 2sin(x) cos(x), where X (2) is the random variable for the 2nd order statistic. (Remember, we only have 2 random variables.) b. Show that fx(1)(x) = 2cos(r) [1 sin(x)], where X(1) is the random variable for the 1st order statistic.Example 2.1.1 Let S = R, and suppose P is a probability measure on R. Define F(x) by F(x) = P((-∞, x]), XER. (2.3) Then (i) F is right continuous,(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.
- 8. Let the random variable X have the pdf 2 x2 fx (x) = exp %3D - V2n 2 Find the mean and the variance of X. Hint: Compute E (X) directly and E (X²) by comparing the integral with the integral representing the variance of a random variable that is N(0,1). i DCO 04 < (X - 5)2 < 38.4).2. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. (a) Find P(X1 = 2 and X2 = 2). (b) Find P(X1 + X2 = 1). (c) Find E(X1 + X2). (d) Find Var(X1 + X2).(11) Let X be a random variable with p.d.f. k 1Let pX(x) be the pmf of a random variable X. Find the cdf F(x) of X and sketch its graph along with that of pX(x) if pX(x)=1/3,x=−1,0,1, zero elsewhereTheorem 11. Let X be a random variable and let g(x) be a non-negative function. Then for r > 0, Eg (X) P[g(X) > r] < Proof.Let X be a continuous random variable with p.d.f. f(x) and distribution function F(x). If Y = X², (a) what is the g(y)? 14/12 (b) If ƒ(x) = -√2/² 2π -∞Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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