1. Complete the proof of Equation (4.4.2), i.e., verify the identity

(d)/(dy)[ ∑k=j^nnk FX^k(y)

Name: 1. Complete the proof of Equation (4.4.2), i.e., verify the identity (d)/(dy)[ ∑k=j^nnk FX^k(y)(1 - FX(y))^n-k] = (n!)/((j-1)!(n-j)!) FX^j-1(y)(1 - FX(y))^n-j fX(y).
Uploaded: 2024-03-07T03:58:35-08:00
Duration: 2 min 21 s
Channel: Hoan Nguyen
Description: 1. Complete the proof of Equation (4.4.2), i.e., verify the identity (d)/(dy)[ ∑k=j^nnk FX^k(y)(1 - FX(y))^n-k] = (n!)/((j-1)!(n-j)!) FX^j-1(y)(1 - FX(y))^n-j fX(y).

Question

1. Complete the proof of Equation (4.4.2), i.e., verify the identity

(d)/(dy)[ ∑k=j^nnk FX^k(y)

Hoan Nguyen · Accepted Answer

Step 1:  We can use the product rule to differentiate the left-hand side of the equation.

Question

1. Complete the proof of Equation (4.4.2), i.e., verify the identity\\ $\frac{d}{dy} \left[ \sum_{k=j}^{n} \binom{n}{k} F_X^k(y)(1 - F_X(y))^{n-k} \right] = \frac{n!}{(j-1)!(n-j)!} F_X^{j-1}(y)(1 - F_X(y))^{n-j} f_X(y). $

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