Question

Let X be a continuous random variable with a PDF of the form

fX(x)={c(1−x),0,if x∈[0,1],otherwise. c= P(X=1/2)= P(X∈{1/k:k integer, k≥2})= P(X≤1/2)= |

Answer #1

Let X be a continuous random variable with a PDF of the form
fX(x)={c(1−x),0,if x∈[0,1],otherwise.
Find the following values.
1. c=
2. P(X=1/2)=
3. P(X∈{1/k:k integer, k≥2})=
4. P(X≤1/2)=

3. Let X be a continuous random variable with PDF
fX(x) = c / x^1/2, 0 < x < 1.
(a) Find the value of c such that fX(x) is indeed a PDF. Is this
PDF bounded?
(b) Determine and sketch the graph of the CDF of X.
(c) Compute each of the following:
(i) P(X > 0.5).
(ii) P(X = 0).
(ii) The median of X.
(ii) The mean of X.

5. Let X be a continuous random variable with PDF
fX(x)= c(2+x), −2 < x < −1,
c(2−x), 1<x<2,
0, elsewhere
(a) Find the value of c such that fX(x) is indeed a PDF.
(b) Determine the CDF of X and sketch its graph.
(c) Find P(X < 1.5).
(d) Find m = π0.5 of X. Is it unique?

Let X be a random variable with pdf given by fX(x) = Cx2(1−x)1(0
< x < 1), where C > 0 and 1(·) is the indicator
function.
(a) Find the value of the constant C such that fX is a valid
pdf.
(b) Find P(1/2 ≤ X < 1).
(c) Find P(X ≤ 1/2).
(d) Find P(X = 1/2).
(e) Find P(1 ≤ X ≤ 2).
(f) Find EX.

2. Let X be a continuous random variable with pdf given by f(x)
= k 6x − x 2 − 8 2 ≤ x ≤ 4; 0 otherwise.
(a) Find k.
(b) Find P(2.4 < X < 3.1).
(c) Determine the cumulative distribution function.
(d) Find the expected value of X.
(e) Find the variance of X

Let X and Y be continuous random variable with joint pdf
f(x,y) = y/144 if 0 < 4x < y < 12 and
0 otherwise
Find Cov (X,Y).

Let X be a continuous random variable with probability density
function (pdf) ?(?) = ??^3, 0 < ? < 2.
(a) Find the constant c.
(b) Find the cumulative distribution function (CDF) of X.
(c) Find P(X < 0.5), and P(X > 1.0).
(d) Find E(X), Var(X) and E(X5 ).

Let X and Y be random variables with the joint pdf
fX,Y(x,y) = 6x, 0 ≤ y ≤ 1−x, 0 ≤ x ≤1.
1. Are X and Y independent? Explain with a picture.
2. Find the marginal pdf fX(x).
3. Find P( Y < 1/8 | X = 1/2 )

4. Consider a continuous random variable X which has pdf fX(x) =
1/7, 0 < x < 7.
(a) Find the values of µ and σ^ 2 . (You may recognize the model
above, and if you do, it is OK to simply write down the answers if
you know them.)
(b) A random sample of size n = 28 is taken from the above
distribution. Find, approximately, IP(3.3 ≤ X ≤ 3.51). Hint: use
the CLT.

a continuous random variable X has a pdf f(x) = cx, for
1<x<4, and zero otherwise.
a. find c
b. find F(x)

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