Lab covers some fundamental derivations related to the normal distribution. The required exercises
borrow many techniques from univariate calculus including u-substitution, change coordinate system.
integration by parts, chain rule. and the fundamental theorem of calculus. This lab could be named "A
of calc problems"
1 Normalizing the normal distribution
Section I walks students through the process of evalunting the famous integral
The process follows:
Note that zisa dummy variable so we can use place of z. Therefore we can write
Shom that A² can be expressed as
1.ii Change the above integral to polar coordinates by applying the transformation
Note that the limits of integration must also change terms radius and angle 0
1.iii Evaluate the integral A² using polar coordinates
Liv Based on part (1.iii), comclude that
1.v Based on part (1.iv), evaluate the following integral:
In this problem you will have to choose the appropriate substitution Hint: z-score
2 Linear transformation of the normal random variable
Section II shows how to compute the probability distribution of linear transformation of X. Let X ~
Níp. of) and define the random variable =aX -6 Note that bis any real number and e is amy positive
real number. Also note that the cdiofris
which does not have closed form solution. i.e. you can't simplify the expression further
Derive the pdfofy by following the process below:
Using the transformation Y aX +6 write Fr(g): terms of Fx(x). This task started below:
Hint: don't express Fy(g) as an integral. Fy can be expressed the form Fx(?)
2.ii Find the pdi of Y by evaluating the derivative
2.iii Based om part (2.ii), identify the probability distribution of y 6
Expectation and variance of the normal random variable
Section III requires students to derive the expected value and variance of normal random variable. Let
X- N(av.o2) and ~N(0,1).
i. Show E|Z] =0 (Hint: u-substitution
ii. Show E\Z4 (Hint: integration by parts)
iii. Compute Var|Z]. (Hint: very easy)
iv. Show E[X) = and Var[X| by applying linearity properties of expectation and variance. To
solve this, think about the relationship between and z.
Some Theory Problems (Part I)
Consider Geometric random variable with pmf
1.1 Notice that
Using the above relation compate E\X2 E[X]. This requires: (i) evaluating the geometric series,
and (ii) computing the second derivative. Simplify your final answer.
Lii Based on part (1.i). find E|X²) and Var[X].
1.ii For -log(1 p). find Eleer (note: log(u) In(u) loe with base e)
Consider Binomail randorn variable x with success probability p € (0. 1) and pmf
Let continuous randorn variable with probability density function fx(x). If and bare real numbers
and =aX then show
Show if X exponential(A) then for any real numbers .P(X>s+t)X >s) =
This shows that the exponential random variable memorgless
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