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If X is  find  and  pnorm. Table II

1. Evaluate  2. Determine the 90th percentile of the distribution, which is N(65, 25). 3. If  2. Let the random variable X have the pdf (a) Find the mean and the variance of X. (b) Find the cdf and hazard function of X.

If Show that the constant c can be selected so that , satisfies the conditions of a normal pdf. If . Show that the graph of a pdf has points of inflection at and .

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 ,

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

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

For Z distributed N(0, 1), it can be shown that see Azzalini (1985). Use this fact to obtain the mgf of the pdf (3.4.20). Next obtain the mean of this pdf.

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).

Establish formula (3.5.11) by a direct multiplication. Formula 3.5.11

Let X and Y have a bivariate normal distribution with parameters