# A univariate Gaussian or normal distributions can be completely determined by its mean and variance. Gaussian distributions can be applied to a large numbers of problems because of the central limit theorem (CLT). The CLT posits that when a large number of independent and identically distributed (i.i.d.) random variables are added, the cumulative distribution function (cdf) of their sum is approximated by the cdf of a normal distribution. Recall the probability density function of the univariate Gaussian with mean u and variance o?, N (1, 02): fx (2) = 1 27102 (2–)*/(20) e Probability review: PDF of Gaussian distribution 2 points possible (graded) In practice, it is not often that you will need to work directly with the probability density function (pdf) of Gaussian variables. Nonetheless, we will make sure we know how to manipulate the (pdf) in the next two problems. The pdf of a Gaussian random variable X is given by n n2(x – 2) fx (2) exp 327 then what is the mean p and variance o2 of X? (Enter your answer in terms of n.) M= 02 Probability review: PDF of Gaussian distribution 1 point possible (graded) Let X~ N (4,02), i.e. the pdf of X is 1 (x – u)? fx (x) exp 027 202 Let Y = 2X. Write down the pdf of the random variable Y. (Your answer should be in terms of y, o and p. Type mu for sigma for o.) fy (y) = Argmax 1 point possible (graded) Let fx (x;j, o?) denote the probability density function of a normally distributed variable X with mean u and variance 02. What value of x maximizes this function? (Enter mu for the mean ji, and sigma-2 for the variance o?.)

A univariate Gaussian or normal distributions can be completely determined by its mean and variance. Gaussian distributions can be applied to a large numbers of problems because of the central…