Stat261

Assignment #3

Neatly hand write your solutions – marks will be assigned for presentation

1. An experiment was conducted to estimate γ, the 90th percentile (0.90) of the lifetime distribution of a new type of transistor. The 90th percentile is the value, γ such that

P(X ≤ γ) = 0.90.

Ten transistors were tested and the observed lifetimes were:

9, 25, 6, 18, 43, 17, 12, 10, 18, 22.

Assuming that the lifetimes follow an exponential distribution with mean θ, find the maximum

likelihood estimate of γ, where γ satisfies, .90 = R γ

0

f(x; θ)dx =

R γ

0

1

θ

exp(−x/θ) dx.

2. Pea plants are classified according to the shape, round (R) or angular (A), and colour, green

(G) or yellow (Y), of the peas they produce. According to genetic theory, the four possible

plant types, RG, RY, AG, AY have probabilities αβ, α(1 − β), (1 − α)β, and (1 − α)(1 − β),

respectively, with different peas being independent of one another.

The following table shows the observed frequencies of the four types in 100 plants examined:

Plant Type RG RY AG AY

Observed frequency 55 21 19 5

Find the MLE’s of α and β, and compute the estimated expected frequencies for each possibility

under the model. You need not show the second derivative conditions.

3. Suppose that X1, X2, …Xn are independent normal variates with the same variance σ

2

, but with

different means,

Xi ∼ N(µbi

, σ2

), for i = 1, 2, …n

where b1, b2, …bn are known constants. Find expressions for the MLE of µ and σ

2

. You need

not show the second derivative conditions.

1

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