Chapter 4, Problem 108SE

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 Deep-Sea 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 = oocyte diameter (µm):

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 σ = .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? [Hint: Write an expression for the corresponding integral, carry the integral operation through to the two components, and then use the fact that each component is a normal pdf.]

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?