• (HW_3_1) Suppose X ~ Normal(m = 23, s = 2). The following table of probabilities was obtained from Excel.

x	P(X<=x)	x	P(X<=x)	x	P(X<=x)
15	0.0000	20	0.0668	25	0.8413
16	0.0002	21	0.1587	26	0.9332
17	0.0013	22	0.3085	27	0.9772
18	0.0062	23	0.5000	28	0.9938
19	0.0228	24	0.6915	29	0.9987




Compute the following probabilities:
(i ) pr(X <= 19);   (ii) pr(X < 19);  (iii) pr(X > 21);
(iv) pr(24 <= X <= 27)

• (HW_3_2) The number of liters of soft serve ice cream sold by an ice cream van driver a day is found to be Normally distributed with a mean of 8.6 liters and a standard deviation of 1.28 liters. The following table of probabilities was obtained from STATA:

P(X<=x)	x
0.1000	6.9596
0.2500	7.7367
0.7500	9.4633
0.9000	10.2404

(i) What is the least amount of soft serve ice cream that is needed so that the driver can satisfy demand on 90% of afternoons?

(ii) What is the inter quartile range for the ice cream sales.
 
(iii) What is the probability that X is greater than 6?

• (HW_3_3) X has a mean of -3 and a standard deviation of 5 and W has a mean of 5 and a standard deviation of 3. Let X and W be independent Normal random variables and let Y = 3X - 3W.

(i) What are the mean and standard deviation of Y?

(ii) What can we say about the shape of the distribution of Y?




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• (HW_3_4) Use Poisson approximation to Binomial distribution to answer the following question:

The performance of a communication network is seriously degraded if more than 0.001% of the transferred bits (0's & 1's) are incorrectly decoded (received). The probability that a single bit is corrupted during transmission is 2x10^-5 and that chance that any particular bit is incorrectly  transmitted is independent of any other bits. Approximately what is the probability that a message of 10⁵ bits is  seriously degraded?