Let a < b be real numbers. let x be a random variable such that it is known that a < = x < = b and for every subinterval of [a; b], the probability that x will belong to that subinterval is proportional to the length of that subinterval. then we say x has the uniform distribution on the interval [a; b] and the probability density function of x is given by: {1/b-a if a< =x< =b 0 otherwiseand we write x- u(a, b)(a) find e[x]. does this agree with your intuition? (b) show var[x]= b^2-a^2/12 hint: b^3- a^3 = (b- a) (b2^+ ab + a^2)(c) find the c. d.f of x.

a)

b)

So first we need to find the second central moment like this:

And now we can find the variance like this:

c)

Step-by-step explanation:

For this case we assume that

And the density function is given by

Part a

In order to find the expected value we need to do the following integral:

And yes agree with the intuition since is the middle value of the interval (a,b)

Part b

For this case we need to use the definition of variance:

So first we need to find the second central moment like this:

And now we can find the variance like this:

Part c

In order to find the cumulative distribution function we just ned to do the following integral:

i dont know i just need points

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range: