An oocyte is a female germ cell involved in reproduction. Based on analyses of a large sample, the article "Reproductive Traits of Pioneer Gastropod Species Colonizing DeepSea Hydrothermal Vents After an Eruption" (Marine Biology, 2011: 181-192) proposed the following mixture of normal distributions as a model for the distribution of X 5 oocyte diameter (mm):

f (x) = pf1(x; μ1, σ) + (1 - p) f2(x; μ2, σ)

where f1 and f2 are normal pdfs. Suggested parameter values were p = .35, μ1 = 4.4, μ2 = 5.0, and θ = .27.

a. What is the expected (i.e. mean) value of oocyte diameter?

b. What is the probability that oocyte diameter is between 4.4 µm and 5.0 μm?

c. What is the probability that oocyte diameter is smaller than its mean value? What does this imply about the shape of the density curve?