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17.1SB2 Statistics 2 Problem sheet 5 – Continuous distributions/Normal distribution 1. Let X ~ Uniform(0, 1) and suppose that Y = g(X) = X3. Identify the inverse mapping g-1 and the range of Y. Using one of the formulae given in lectures (Part III , Section 5), calculate the pdf of Y. 2. Let X be a continuous random variable whose pdf is fX(x) = 2x, 0 < x < 1. i) Sketch the graph of fX(x). Would you expect the mean of X to be greater or less than 0.5? ii) By evaluating appropriate integrals, calculate E(X), E(X2) and, hence, the variance of X. 3. Let X be a random variable whose distribution is N(100, 100). Use tables of the Normal cumulative distribution function to calculate the following probabilities: i) P(X > 115); ii) P(X < 92); iii) P(95 < X < 120). (Hint: First transform X to a standard Normal random variable Z using the transformation discussed in lectures.) 4. The gestation period (measured in days) for human births can be taken as normal N(, 2) with = 266 and = 16. i) Calculate the probability that a gestation period lasts for more than 290 days. ii) What is the probability of a gestation period lasting between 250 and 282 days? 5. A medical trial was conducted to investigate whether a new drug extended the life of patients with a certain disease. The drug was given to 38 patients and their survival times (in months) were recorded. These were (in ascending order): 1 21 39 50 1 22 40 50 5 25 41 54 9 25 41 54 10 25 43 59 13 26 44 14 27 44 17 29 45 18 36 46 18 38 46 19 39 49 i) Calculate the median and the quartiles Q1, and Q3 for these data. ii) Construct a stem-and-leaf diagram for the data. Does the plot support the suggestion that the distribution of survival times is Normal?