Question

1. The density function of b is given by kx(1 - x) f(x) = { for 0 < x 51, elsewhere. (a) Find k and graph the density function. (b) Find P(1/4 < ſ < 1/2). (c) Find P(-1/2 sã < 1/4). (d) Find the CDF and graph it. (e) Find E( ), E(52), and V(5).

Answer 1

1. Suppose a random sample o has density function prask sc(1-) for oqsi, o elsewhere. (a) To find: k and graph of density function. We know that area under density care Ps 1. Let us consider this fact to find the value of ka

That is, k =(1-x) dx = 1 dx = 1 sa jee -zºdx=1 >> * [jike-fatahes Tech I xd nti » 6-1)-6-0) = % = = = » * =6. Thus, density function of ã is given by CGali-se), os aci, f(0) = 0 , otherwise. Now, to sketch consider the table below. (For a rough idea.) 500 0.2 0.4 0.5 10.6 To 1pbo) 0 | 0.96 1.44 .5 11.44 10.96

Hence , apporocimately the density Cuove lodos libe the below figure. Probability Density Carve fo fea ] I (6) To find: PC/4 cã</2) consider, PC/4 C5 (4)= | 6l1-a) doc 6 (1-) de X2 8x3) 16X2 64x3 PGLĩ4) -

(C) To find: p(-/2 cã sva). Since the given function is a density function 1.e., continuous distribution over the 'Xange Co, 3, inequalities sorc are always inclusive. Consider PC-13 Sã¢%4)= plosã</a). Since range of i.e., 44 plocast) = |6x1-x)da Curve below '' is does not exist, bence the probability is zero. 16x2 64x3 Plocãcla)= a (d) To find: CDF (cumulative Distribution function) and graph. Consider CDF, a FC60)=P(x <=)= 1 60261-3) doc

F%) = (x = = () . 3922-20. Hence, the CDF s 300-26, ocasi o , otherwise. For beloco. sketching conside the table FC) o 0.104 0.352 0.5 0.698 0.894 | I Cumulative Distribution Ceuve ↑ at Harjost i io ' 0 0.2 0.4 0.6 0.8 1