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Compute the probability

Let If X is N(1, 4), compute the probability If X is N(75, 25), find the conditional probability that X is greater than 80 given that X is greater than 77. See Exercise 2.3.12. Exercise 2.3.12 Let and denote, respectively, the pdf and the cdf of the random variable X. The conditional pdf of ,

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Region of integration

Consider the family of pdfs indexed by the parameter , given by where φ(x) and Φ(x) are respectively the pdf and cdf of a standard normal distribution. (a) Clearly  fo all x. Show that the pdf integrates to 1 over . Next sketch the region of integration and then combine the integrands and use the polar coordinate transformation

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Domain values

Plot the pdfs of the random variables defined in parts (a)–(d) of the last exercise. Obtain an overlay plot of all four pdfs also. In R the domain values of the pdfs can easily be obtained by using the seq command. For instance, the command returns a vector of values between −6 and 6 in

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Skewed normal family

Note that  is the  pdf for  The pdfs are left skewed for α < 0=”” and=”” right=”” skewed=”” for=”” α=””> 0. Using R, verify this by plotting the pdfs for α = −3, −2, −1, 1, 2, 3. Here’s the code for α = −3: This family is called the skewed normal family; see Azzalini (1985).

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