7 Leaves are divided into four different types starchy green

7. Leaves are divided into four different types: starchy green, sugary white, starchy white, and sugary green-According to genetic theory, the es occur with probability (0+2. and (1-0), respectively, where o 0 I. Suppose we have n leaves. Then the number of starchy green leaves is modeled by a random variable Ni with a Bin(n, distribution, where p 10- 2), and the number of sugary white leaves is madeledby a random variahle N2 with a Binin, distribution, where p 16. Consider the fallowing two estimatars for T- N-2 and T- N. You previously shawed that both T and T are unbiased estimators for 0. Which estimator would you prefer? Mativate your

Solution

We know that T1 and T2 are unbiased estimators of theta

To determine the better estimator, we test the MSE for both cases.

Since the estimators are unbiased, they must be equal to their variances.

Both N1 and N2 are binomally distributed, so Var(N1) = np₁ (1 - p₁)

= n [ (t+2)/4] [ 1 - (t+2)/4 ]

Similarly,

Var (N2) = nt (1-t) /4

We are given that,

T1 = (4/n)N₁ - 2 and T2 = (4/n)N₂

Thus, it follows from the above equations,

Var (T1) = (16 / n² ) Var(N₁) = (1/n) (4 - t² )

Var (T2) = (16 / n² ) Var(N₁) = (1/n) t(4 - t )

Since, for all values of t.

(4 - t² ) > t(4 - t )

T2 is a better estimator (less biased) than T1

Hope this helps.