Homework Answers
The pdf of random variable X is shown in Figure P2.10. The numbers in parentheses indicate...
The pdf of random variable X is shown in Figure P2.10.
The numbers in parentheses indicate area. (a) Compute the value
of A. (b) Compute P[2 <X< 4].
The answers should be: (a) 2.5053 (b) 0.4174
fx(x) 0 1 2 4. Figure P2.10 pdf of a Mixed RV
4) (20 pts) Let X be a RV with the following PDF: fx(x) = že=fal for...
4) (20 pts) Let X be a RV with the following PDF: fx(x) = že=fal for all x. Let Y = X?. (a) Compute E[X]. (b) Find the PDF of Y, fy(y). (c) Compute E[Y].
Question # A.4 (a) Given that probability density function (pdf of a random variable (RV), x is a...
Question # A.4 (a) Given that probability density function (pdf of a random variable (RV), x is as follows: Px(x)-axexp(-ax) x 20 otherwise where α is a constant. Suppose y = log(x) and y is monotonic in the given range of X. Determine: (i) pdf of y; (ii) valid range of y; and, (iii) expected value of y. Answer hint:J exp(y) (b) Given that, the pdf, namely, fx(x) of a RV, x is uniformly distributed in the range (-t/2, +...
continuous RV [4/5] X has PDF: f(x) = ae-|x| Compute the value of the constant a...
continuous RV [4/5] X has PDF: f(x) = ae-|x| Compute the value of the constant a Compute E[X]
The joint pdf for rv X, Y is given as follows: if 1 ? x,y ?...
The joint pdf for rv X, Y is given as follows:
if
1 ? x,y ? 2 and it is zero else.
Find:
(a) The value of c
(b) E(X)
(c) E(Y)
(d) E(X|Y)
(e) Var(X|Y)
(f) The MMSEE of eX given Y , E(eX|Y )
(g) Are X and Y independent?
fx,y(x, y) = c(2²/y)
3.17 A PDF for a continuous random varaiable X is defined by C 0<x<2 2C4<< 6...
3.17 A PDF for a continuous random varaiable X is defined by C 0<x<2 2C4<< 6 fx(x) = 3 C 7<<<9 0 otherwise where C is a constant. (a) Find the numerical value of C. (b) Compute Pr[1 < X < 8). (c) Find the value of M for which "fx(s)de = [fx (a)dr = 1 J-00 Mis known as the median of the random variable.
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist...
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X). b) Show that Z=-2ln(Y) has a Gamma dist. & derive it. 4. X_i ~ cont with pdf f_i(x) and CDF F_i(x), i=1, 2, ..., k. all independent. Define Y_i=F_i(X_i), i=1, ..., k. Derive the distribution of U=-2ln[Y_1.Y_2...Y_k].
Problem 4: Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y: xe-(+) ;x&...
Problem 4: Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y: xe-(+) ;x>0;y>0 0 ; elsewhere fx y(x,y)- (a) Explain whether the lifetimes of two components are independent based on probability. (b) Compute the probability that the lifetime (X) exceeds 3.5 (c) Compute the probability that the lifetime of at least one component exceeds 3.5. (d) Compute the marginal pdf of X
Problem 4: Two components of a minicomputer have the following...
Problem 4: Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y: xe-(+) ;x&...
Problem 4: Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y: xe-(+) ;x>0;y>0 0 ; elsewhere fx y(x,y)- (a) Explain whether the lifetimes of two components are independent based on probability. (b) Compute the probability that the lifetime (X) exceeds 3.5 (c) Compute the probability that the lifetime of at least one component exceeds 3.5. (d) Compute the marginal pdf of X
Problem 4: Two components of a minicomputer have the following...
compute the pdf of the failure 9Compute the pdf of the failure time X if the...
compute the pdf of the failure
9Compute the pdf of the failure time X if the conditional failure rate a() is as shown in Figure P2.39. alt) 10 Figure P2.39 Failure rate a(t) in problem 2.39
The pdf of random variable X is shown in Figure P2.10.
The numbers in parentheses indicate area. (a) Compute the value
of A. (b) Compute P[2 <X< 4].
The answers should be: (a) 2.5053 (b) 0.4174
fx(x) 0 1 2 4. Figure P2.10 pdf of a Mixed RV
4) (20 pts) Let X be a RV with the following PDF: fx(x) = že=fal for all x. Let Y = X?. (a) Compute E[X]. (b) Find the PDF of Y, fy(y). (c) Compute E[Y].
Question # A.4 (a) Given that probability density function (pdf of a random variable (RV), x is as follows: Px(x)-axexp(-ax) x 20 otherwise where α is a constant. Suppose y = log(x) and y is monotonic in the given range of X. Determine: (i) pdf of y; (ii) valid range of y; and, (iii) expected value of y. Answer hint:J exp(y) (b) Given that, the pdf, namely, fx(x) of a RV, x is uniformly distributed in the range (-t/2, +...
continuous RV [4/5] X has PDF: f(x) = ae-|x| Compute the value of the constant a Compute E[X]
The joint pdf for rv X, Y is given as follows:
if
1 ? x,y ? 2 and it is zero else.
Find:
(a) The value of c
(b) E(X)
(c) E(Y)
(d) E(X|Y)
(e) Var(X|Y)
(f) The MMSEE of eX given Y , E(eX|Y )
(g) Are X and Y independent?
fx,y(x, y) = c(2²/y)
3.17 A PDF for a continuous random varaiable X is defined by C 0<x<2 2C4<< 6 fx(x) = 3 C 7<<<9 0 otherwise where C is a constant. (a) Find the numerical value of C. (b) Compute Pr[1 < X < 8). (c) Find the value of M for which "fx(s)de = [fx (a)dr = 1 J-00 Mis known as the median of the random variable.
Problem 4: Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y: xe-(+) ;x>0;y>0 0 ; elsewhere fx y(x,y)- (a) Explain whether the lifetimes of two components are independent based on probability. (b) Compute the probability that the lifetime (X) exceeds 3.5 (c) Compute the probability that the lifetime of at least one component exceeds 3.5. (d) Compute the marginal pdf of X
Problem 4: Two components of a minicomputer have the following...
Problem 4: Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y: xe-(+) ;x>0;y>0 0 ; elsewhere fx y(x,y)- (a) Explain whether the lifetimes of two components are independent based on probability. (b) Compute the probability that the lifetime (X) exceeds 3.5 (c) Compute the probability that the lifetime of at least one component exceeds 3.5. (d) Compute the marginal pdf of X
Problem 4: Two components of a minicomputer have the following...
compute the pdf of the failure
9Compute the pdf of the failure time X if the conditional failure rate a() is as shown in Figure P2.39. alt) 10 Figure P2.39 Failure rate a(t) in problem 2.39
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