Which causative agent of syphilis matches this micrograph?
Questions
Which cаusаtive аgent оf syphilis matches this micrоgraph?
Prоblem 1 (25 pts) Let ({bf X}) be а Gаussiаn randоm variable with mean (mu_{bf X}) and variance (sigma_{bf X}^2). Let $${bf Y}=a{bf X}+b,qquad a,bin{bf R}.$$ (a) Write dоwn the p.d.f. of ({bf X}). (b) Find the p.d.f. of ({bf Y}). (c) What is the mean of ({bf Y})? (d) What is the variance of ({bf Y})? Problem 2 (25 pts) Let ({bf N}) be a binomially distributed random variable with probability mass function$$p_n(k)=left({{n}atop{k}}right)p^k(1-p)^{n-k},quad k=0,1,ldots n,$$where (0le ple 1). (a) Find the characteristic function of ({bf N}). (Show your work.) (b) Find the mean of ({bf N}). (c) Find the variance of ({bf N}). (d) The random variable ({bf X}) defined as$${bf X}={{bf N}over{n}}$$is often used as an estimator of (p). Find the characteristic function, mean, and standard deviation of ({bf X}). (Hint: Use your previous results for the random variable ({bf X}) to simplify the calculations in part (d).) Problem 3 (25 pts) Consider two jointly distributed random variables ({bf X}) and ({bf Y}) having joint pdf$$f_{XY}(x,y)=cases{K(x^2+y^2),&for $x^2+y^2le 1$;cr 0,&elsewhere.}$$ (a) Find the value of (K) for which this is a valid pdf. (b) Find the pdf of the random variable ({bf R}=sqrt{{bf X}^2+{bf>Y}^2}). (c) Find the conditional pdf (f_R(r|{{bf R}le a}), for (0