An oocyte is a female germ cell involved in reproduction.Based on analyses of a large

Chapter 4, Problem 108

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An oocyte is a female germ cell involved in reproduction.Based on analyses of a large sample, the articleReproductive Traits of Pioneer Gastropod SpeciesColonizing DeepSea Hydrothermal Vents After anEruption (Marine Biology, 2011: 181192) proposed the following mixture of normal distributions as a model for the distribution of X = oocyte diameter (mm):

\(f(x)=p f_{1}\left(x ; \mu_{1}, \sigma\right)+(1-p) f_{2}\left(x ; \mu_{2}, \sigma\right)\)

where f1 and f2 are normal pdfs. Suggested parameter values were p = .35, \(\mu_{1}=4.4, \mu_{2}=5.0, \text { and } \sigma=.27\).

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

b. What is the probability that oocyte diameter is between 4.4 \(\mu\) and 5.0 \(\mu\)? [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 normalpdf.]

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?