Question

A law firm specializing in corporate law has 4 lawyers. Because of various reasons, on any given day, the probabilities that each lawyer will be attending a trial is:  0.90, 0.90, 0.85, 0.80. Note that a lawyer attending a trial is independent of any other lawyer attending a trial

a.  Find the probability distribution for the number of lawyers who has a trial on a given day.

b. Find the expected number of lawyers attending to a trial on a given day.

c. Find the standard deviation of the number of lawyers attending to a trial on a given day. Round your answer to four decimal points.

Answer 1

a) The probability distribution of the number of lawyers who has a trial on a given day is computed here as:

P(X = 0) = (1 - 0.9)2*(1 - 0.85)*(1 - 0.8) = 0.0003
P(X = 1) = 0.9*(1 - 0.9)*(1 - 0.85)*(1 - 0.8)*2 + (1 - 0.9)2*0.85*(1 - 0.8) + (1 - 0.9)2*(1 - 0.85)*0.8 = 0.0083
P(X = 2) = 0.92*(1 - 0.85)*(1 - 0.8) + 2*0.9*0.85*(1 - 0.9)*(1 - 0.8) + 2*0.9*0.8*(1 - 0.9)*(1 - 0.85) + 0.85*0.8*(1 - 0.9)2
P(X = 2) = 0.0833
P(X = 3) = 0.92*0.85*(1 - 0.8) + 0.92*0.8*(1 - 0.85) + 0.9*0.85*0.8*(1 - 0.9)*2 = 0.3573
P(X = 4) = 0.92*0.85*0.8 = 0.5508

This is the required PDF for X here:
P(X = 0) = 0.0003
P(X = 1) = 0.0069
P(X = 2) = 0.0833
P(X = 3) = 0.3573
P(X = 4) = 0.5508

b) The expected value of X here is obtained here as:

X	P(X = x)	xP(X = x)	x^2*P(X = x)
0	0.0003	0	0
1	0.0083	0.0083	0.0083
2	0.0833	0.1666	0.3332
3	0.3573	1.0719	3.2157
4	0.5508	2.2032	8.8128
	1	3.45	12.37

Therefore 3.45 is the required expected value of X here.

c) The standard deviation here is computed as:

Therefore 0.6837 is the required standard deviation here.