Question

Question 1

Question 1) (15) Suppose the table below represents the relative frequency of the number of defective items produced per day for a sample 25 days. Number of Defective Items (X) Number ofdays Pr(X-i) 距1 Pr(X =x) 4 a) Fill in the Probability distribution of X. Sketch a graph illustrating the p.d. (2.5) b) Fil in the cumulative distribution of X. Sketch a graph illustrating the c.d. (2.5) c) Calculate the expected value of X. Explain the meaning of the expected value in this particular example. 2 sentences. d) Calculate the Variance of X. Explain the meaning of the variance in this particular example. 2 sentences.

Answer 1

Number of Defective Items (x)	Number of Days	Pr(X=xi)	$\small \Sigma _{5}^{i}$ Pr( X=xi)
0	7	7/25 = 0.28	0.28
1	9	9/25 = 0.36	0.28+0.36=0.64
2	5	5/25 = 0.20	0.64+0.20=0.84
3	3	3/25 = 0.12	0.84+0.12=0.96
4	1	1/25 = 0.04	0.96+0.04=1

c) E(X) = $\small \Sigma _{5}^{i} x _{i}.pr(x_{i})$ = 0(.28) + 1(.36) + 2(.20) + 3(.12) + 4(.04) = 1.28

This means on an average, 1.28 defective items are produced per day.

d) Var(X) = E(X²) - [E(X)]²

E(X²) = $\small \Sigma _{5}^{i} x^{2} _{i}.pr(x_{i})$ = 0(.28) + 1(.36) + 4(.20) + 9(.12) + 16(.04) = 2.88

and  [E(X)]² = 1.6384 therefore Var(X) = 2.88-1.6834 =1.2416

The spread of the data from the conditional expectation or mean is 1.2416