Question

The probability of a new drug working in a patient in a clinical trial is modeled as P ∼ Unif([0, 1]). The drug is then given to patients one at a time, and you observe that the seventh patient is the first patient in which the drug works. That is, if X is a random variable denoting the first patient in which the drug worked, then [X|P] ∼ Geom(P), and you observed X = 7. Conditioned on this information that X = 7, what is the distribution of P?

Answer 1

el 8- Ocp4 Here xfp r Geometric (P). so, p(x=r|p) = (1-P) 12,221,2,3.0 Pwulo, o fep) = 1; O CPC = 0; elsewher so Joint Density f(x=r, b). = (1-P) : 0-1,2.C g(+ x=0) Plx=r) D) ffp l' P(x=2/6) foi ap define, B(a, b) = f at (1-xo da. Beta - Madrb (a-1)! (b-1) fimetion Mato) [a+6)! pl Plxzx|p) Hea dp - ST12-D,p_dp a=2, b= l ri S (1-p7! p dp - B(2,8).

Here . g (plx=r) (1-3) . - B (2,8) Þ r Gengar from. There given that rat. gp|x=7) = + (1-) B/2,7) B/2.70 - 1(2) 17) - 1 6! osana 79) - 8 8x7x4, 1965|x=7a 576 Þ (1-5) 6. Jo<b<l.