Question

The humidity level, X, in Tropicana Field in Tampa at the time of a night game ranges from 40% to 100%. (.40 to 1.00) The random variable x has cumulative distribution F with the following definition: 40 Flyle 5x-.4)(2-x) for 4 sx< 1.0, F(x)=1 for x 2 1.0. F(x) = 0 for x <.40, F(x)= 3 a) What is the probability density function for X for.4 SXS 1.0? 5(2-x)-5(x-0.4) 1-5(x-.4)(2-x) 3 dnorm(x,2,.4 -5(2-x)+5(x-0.4) b) What is the probability that X>0.5? c) What is the probability X< 0.5? ? d) What is the probability that x > 0.6? e) What is the probability that x > 0.8? f) What is the probability that x > 0.8 given the x > 0.6? g) Calculate the 60th percentile of X. h) What is the expected value of X? i) What is the expected value of x2? j) What is the variance of X? k) What is the probability that X is more than 0.1 above its expected value?

Answer 1

Here n. 4) (2-2) f(a) = 0.41n (1.0 7 o otherwise @ density function, tla (say) Probablity = { [FL] = 4(2-4) £4) +(2-x)3 - hl2-a) - tm-4) = 5(2-2) -5(n--4) © P(x>0.5) = 1 fins de 0.5 * *2-)=6–)an dn nant.4 3 os da =ş 1 (2.4 - 2n) dx 30.5 Ş[ 241-2 12205 2.4x0.5 -075) = J x 0.45 = 0.75 P[x<0.5) - 1-PCX>05] - 1-0.75 @ = 0.25

P[X 70.6] +(x) da I 0.6 2.4 - 2x) da o why why why 12 - { [2.48-aug lason - 0.53 p(x > 0.8] - I ten an 6B a 0. 20 A p[x>0.8xx06] a P[*>0.8. n xyoo] P(x706) APEX 70.8I 0.4 0.6 0.8 plx yo.6] 0. 20 0.53 0.38 ( up to two decimal place. g Let the both percentile is k. then, k tens an = 0.60 0.4 - 2 (2.4x-x-) / 20.60 10.4 2.4K- 0.96 - 2 + 0.16 0.36 K²_2.44 +1.16 20 K - 2.4+ √5.76-4.64

K = 1.73 on, k= 0.67 : 60th percentile is a 0.67 [ since 1.73 is out of nange] 6 Elns = Intl dan In20.4 = 5 (2.40 - 2x2) de Czym } ] = { [ 1.008 – 0.624] = 0.64 a E(2-) = f n t (m) an - § 5 (2:44= 2x4) dm = { [ 243 - 2304 - { [0.7488 – 0.3959044 - 0.58816 ③ ranck) - € (m2) - (2) 22 I 0.17856 04 064 we have to find Prorex (0.64] gulay da = teman -[2.42-]/004 -0.544 OY