5. The mgf of a discrete random variable X is given by 1 1 M(t)=ce'|- (2-e²)² (3-e²)²] (a) Find the value of e, the expectation E[X(2X-5)]. t< In 2
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![5. The mgf of a discrete random variable X is given by
M(t)= ce
1
(2-e¹)²
+
1
(3-e¹)²
t< In 2
(a) Find the value of e, the expectation E[X(2X-5)].
(b) Find the pmf of X and P[X≥ 3].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6d69502-ffbd-48d7-85dd-a2e3610b14b4%2Fcd9d334c-8439-498f-bad2-56e512c19bf7%2F6p9gv6r_processed.jpeg&w=3840&q=75)
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- (24) Let X be a random variable with p.d.f. Ze-2 0Let Y be the present value random variable of a 10-year term annuity-due of 1 issued at an individual (35). Calculate its actuarial present value if (a) Mortality follows De Moivre's Law with w = 100. (b) 8 = 0.05.Let X be a random variable taking positive values and assume that E(X) exists. Which of the following statements are true? • (i) E(X*) > (E(X))*. • (ii) E(1/X²) > 1/E(X²). • (ii) E(e-3X) > e°Q2/(a) Let Xbe a random variable with p.d.f. 1b) A continuous random variable X has the p.d.f f(x) = {A(2 – x)(2 + x), 0 < x < 2, Find (i) the value of A, (ii)P(X <1)(iii) P(1 < X <2). l0, otherwis ------------The diameter of a round rock in a bucket, can be messured in mm and considered a random variable X in f(x) f(x) = k(x-x4) if 0 ≤ x ≤ 1f(x) = 0 otherwise. Find the expectation E(4X+ 3) and variance V(4X+3)4. Let X be a positive random variable (i.e. P(X 1/E(X) (b) E(-log(X)) > -log(E(X)) (c) E(log(1/X))> log(1/E(X)) (d) E(X³) > (E(X))³1 (20') The discrete type random variable X has PMF Px(1)= { c(5- ) for a= 1,2, 3, 4 0, otherwise. (a) Determine the value of c. (b) Find E[X] and Var[X]. (c) Find the probability P[X > 2].5. Let X be a positive random variable; i.e., P(X - log(EX) (b) E[log(1/X)] > log[1/EX] (c) E (X³) > (EX)³My Find the standard deviation of the discret probability dist x P(x) xp(x)) (x-M)² | (x-mipul 2 σ = √(x-M)² pxx) ટે О 0.21 10*0.21=0 /10-112 =. M= Exp(x) 1 0.30 1*0.3 = 0.3 2 0=49 EXP(X) =M3) Assume that a random variable X has moment generating function as follows: e' + e²¹ M(t)=4-2e¹ ∞ Calculate the mean and variance of X. for tRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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