tutorial 1 - From the name of the random variable what are...

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2012/13 Semester 2 Tutorial 1 ST3239 1. Suppose there are N elements in a population. The variable of interest for the i th element is denoted by Y i , i = 1, 2, ….., N. (a) Define the average and variance of the variable Y . (b) If an element is selected randomly from the population. Let random variable X be the value attached to the selected element. Find the probability distribution of X . Also, find the mean and variance of X . (c) Compare the results obtained in (a) with the ones in (b). What have you observed? (d) Suppose the probability distribution function of X is given by x 1.2 2.6 3.7 5.9 6 P( X = x ) 0.2 0.1 0.4 0.05 0.25 How do you link this probability distribution function to a population of N elements? (e) Suppose the variable Y i is defined as ܻ ൌ ቄ 1 if the ݅th element has a certain attribute, 0 otherwise What are the forms of average and variance defined in (a)? (f) What is the name of the random variable defined in (b)?
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Unformatted text preview: From the name of the random variable, what are its mean and variance? 2. Suppose there are N people in a population. From each person two variables, X and Y , are measured. Define ܺ ௜ ൌ ቄ 1 if the ݅th person is a female otherwise Suppose Y i is the income of the i th person. Use the two variables to define the average and variance of income of female. 3. Suppose X and Y are random variables with means ߤ ௑ and ߤ ௒ and variances ߪ ௑ ଶ and ߪ ௒ ଶ , respectively. (a) Is ܧ ቀ ௒ ௑ ቁ ൌ ఓ ೊ ఓ ೉ true? (b) Is ܧ ቀ ௒ ௑ ቁ ൌ ఓ ೊ ఓ ೉ true if X and Y are uncorrelated? (c) Is ܧ ቀ ௒ ௑ ቁ ൌ ఓ ೊ ఓ ೉ true if X and Y are independent? (d) How do we derive ܧ ቀ ௒ ௑ ቁ ? (e) How do we derive ܸ ቀ ௒ ௑ ቁ ?...
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