Rh System. Rh antigens are transmembrane proteins with loops exposed at the surf

Question

Rh System. Rh antigens are transmembrane proteins with loops exposed at the surface of red blood cells. They appear to be used for the transport of carbon dioxide and/or ammonia across the plasma membrane. They are named for the rhesus monkey in which they were first discovered. There are a number of different Rh antigens. Red blood cells that are Rh positive express the antigen designated as D. About 15% of the population do not have RhD antigens and thus are Rh negative. The major importance of the Rh system for human health is to avoid the danger of RhD incompatibility between a mother and her fetus. (a) From the general population 8 people are randomly selected and checked for their Rh factor. Let X be the number of Rh negative among the eight selected. Find P(X = 2). (b) In a group of 16 patients, three members are Rh negative. Eight patients are selected at random. Let Y be the number of Rh negative among the eight selected. Find P(Y = 2). (c) From the general population subjects are randomly selected and checked for their Rh factor. Let Z be the number of Rh positive subjects before the first Rh negative subject is selected. Find P(Z = 2). d) Identify the distributions of the random variables in (a), (b), and (c). (e) What are the expectations and variances for the random variables in (a), (b), and (c)?

Explanation / Answer

(a)
p = 0.15
P(X = 2) = 8C2 * p^2 * (1-p)^6 = 8C2 * 0.15^2 * 0.85^6 = 0.2376

(b)
p = 3/16 = 0.1875
P(Y = 2) = 8C2 * p^2 * (1-p)^6 = 8C2 * 0.1875^2 * 0.8125^6 = 0.2832

(c)
Here the first subject wil lnot have Rh negative where as second subject will have
Hence P(Z = 2) = 0.85*0.15 = 0.1275

(d)
(a) Binomial distribution
(b) Binomial distribution
(c) Bernoulli distribution

(e)
(a) E = np = 8*0.15 = 1.2, Var = sqrt(npq) = sqrt(8*0.15*0.85) = 1.01

(b) E = 8*0.1875 = 1.5, Var = sqrt(npq) = sqrt(8*0.1875*0.8125) = 1.104

(c) E = p = 0.15, Var = pq = 0.15*0.85 = 0.1275

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