![Suppose 6 numbers are generated by a computer, each uniform on the interval (0, 1). Let Y be the random variable representing the smallest of the numbers. (a) Show that the probability density of Y is given by py (t) -61-t)5, 0t <1 [51 Hint: The probability density for the r-th largest random variable can be derived using the Beta distribution by letting a = r and ?-n-r +1. (b) What is the probability that the smallest number is less than 0.5? 15 (c) What is the expected value and variance of each of the uniform random variables?12] [41 (d) What is the expected value and variance of the random variable y](http://img.homeworklib.com/questions/44176680-a529-11eb-892f-53303e7bb0f3.png?x-oss-process=image/resize,w_560)
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Suppose 6 numbers are generated by a computer, each uniform on the interval (0, 1). Let...
4.60 The sum of two uniform random numbers. Generate two random numbers between 0 and 1...
4.60 The sum of two uniform random numbers. Generate two random numbers between 0 and 1 and take Y to be their sum. Then Y is a continuous random variable that can take any value between 0 and 2. The density curve of Y is the triangle shown in Figure 4.12. (a) Verify by geometry that the area under this curve is 1. (b) What is the probability that Y is less than 1? [Sketch the density curve, shade the...
1) 2) 3) 4) 5) Suppose that X is a uniform random variable on the interval...
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Suppose that X is a uniform random variable on the interval (0, 1) and let Y = 1/X. a. Give the smallest interval in which Y is guaranteed to be. Enter -Inf or Inf for – or o. Interval:( b. Compute the probability density function of Y on this interval. fy(y) = Suppose that X ~ Bin(4, 1/3). Find the probability mass function of Y = (X – 2)2. a. List all possible values that...
1. Let X be a random variable with variance ? > 0 and fx as a...
1. Let X be a random variable with variance ? > 0 and fx as a probability density function (pdf). The pdf is positive for all real numbers, that is fx(x) > 0. for all r ER Furthermore, the pdf fx is symmetric around zero, that is fx(x) = fx(-1), for all r ER Let y be the random variable given by Y = 4X2 +6X + with a,b,c E R. (i) For which values of a, b, and care...
1. A point P is chosen with a uniform probability distribution around a circle of radius...
1. A point P is chosen with a uniform probability distribution around a circle of radius r Let Z be a random variable that measures the absolute value of the distance of P from the y-axis (a) What is the mean and the variance of Z? (Hint, define an appropriately normalized uniform probability density function for the angle 0 describing the polar angle of the position P on the circle.) (b) Does your answer for the mean make sense? (c)...
Suppose that you specify that the range is to be all numbers between 0-4. Call the...
Suppose that you specify that the range is to be all numbers between 0-4. Call the number generated y. Then the density curve of the random variable y has constant height between 0-4 and height 0 elsewhere. a. what is the height of the density curve = 0.25 b. use your graph from a and the fact that probability is area under the curve to dfind P(Y<1.8) c. Find P (0.6<Y<1.8) d. Find P(Y>0.9)
lo (P15) Suppose X is a random variable with the uniform distribution over the interval (1.2)...
lo (P15) Suppose X is a random variable with the uniform distribution over the interval (1.2) and Y = X4 (a) Compute P[Y St] as a function of t. You need to distinguish three different cases. (b) Find the probability density function of Y and use it to compute EY).
Let Y1, Y2, ..., Yn be independent random variables each having uniform distribution on the interval...
Let Y1, Y2, ..., Yn be independent random variables
each having uniform distribution on the interval (0, θ).
(a) Find the distribution of Y(n) and find its expected
value.
(b) Find the joint density function of Y(i) and Y(j) where 1 ≤ i
< j ≤ n. Hence
find Cov(Y(i)
, Y(j)).
(c) Find var(Y(j) − Y(i)).
Let Yİ, Ya, , Yn be independent random variables each having uniform distribu- tion on the interval (0, 6) (a) Find the distribution...
I. Let the random variable y have an uniform distribution with minimum value θ = 0...
I. Let the random variable y have an uniform distribution with minimum value θ = 0 and maximum value θ2-1 and let the random variable U have the form aY +b, where a and b are both constants and a > 0. (a) Using the transformation method, find the probability density function for the random variable U when a 2 and b-4. What distribution does the random variable U have? (b) Using the transformation method, find the probability density function...
S2-R be a random variable on a probability space (LF, P) with the uniform distribution on [1-1,T+름 . Does there exist a random variable Y : Ω → R For each n E N, let Yn such that Y,,-, Y almost sure...
S2-R be a random variable on a probability space (LF, P) with the uniform distribution on [1-1,T+름 . Does there exist a random variable Y : Ω → R For each n E N, let Yn such that Y,,-, Y almost surely as n-> oo?
S2-R be a random variable on a probability space (LF, P) with the uniform distribution on [1-1,T+름 . Does there exist a random variable Y : Ω → R For each n E N, let...
Let X be a random variable with the probability density function f(x)= x^3/4 for an interval...
Let X be a random variable with the probability density function f(x)= x^3/4 for an interval 0<x<2 (a) What is the support of X? (b) Letting S be the support of X, pick two numbers a, b e S and compute Pa<x<b). Draw a graph that shows an area under the curve y = f() that is equal to this probability. (c) What is Fx (2)? Draw a good graph of y=Fx (I). (d) What is EX? (e) What is...
4.60 The sum of two uniform random numbers. Generate two random numbers between 0 and 1 and take Y to be their sum. Then Y is a continuous random variable that can take any value between 0 and 2. The density curve of Y is the triangle shown in Figure 4.12. (a) Verify by geometry that the area under this curve is 1. (b) What is the probability that Y is less than 1? [Sketch the density curve, shade the...
1)
2)
3)
4)
5)
Suppose that X is a uniform random variable on the interval (0, 1) and let Y = 1/X. a. Give the smallest interval in which Y is guaranteed to be. Enter -Inf or Inf for – or o. Interval:( b. Compute the probability density function of Y on this interval. fy(y) = Suppose that X ~ Bin(4, 1/3). Find the probability mass function of Y = (X – 2)2. a. List all possible values that...
1. Let X be a random variable with variance ? > 0 and fx as a probability density function (pdf). The pdf is positive for all real numbers, that is fx(x) > 0. for all r ER Furthermore, the pdf fx is symmetric around zero, that is fx(x) = fx(-1), for all r ER Let y be the random variable given by Y = 4X2 +6X + with a,b,c E R. (i) For which values of a, b, and care...
1. A point P is chosen with a uniform probability distribution around a circle of radius r Let Z be a random variable that measures the absolute value of the distance of P from the y-axis (a) What is the mean and the variance of Z? (Hint, define an appropriately normalized uniform probability density function for the angle 0 describing the polar angle of the position P on the circle.) (b) Does your answer for the mean make sense? (c)...
lo (P15) Suppose X is a random variable with the uniform distribution over the interval (1.2) and Y = X4 (a) Compute P[Y St] as a function of t. You need to distinguish three different cases. (b) Find the probability density function of Y and use it to compute EY).
Let Y1, Y2, ..., Yn be independent random variables
each having uniform distribution on the interval (0, θ).
(a) Find the distribution of Y(n) and find its expected
value.
(b) Find the joint density function of Y(i) and Y(j) where 1 ≤ i
< j ≤ n. Hence
find Cov(Y(i)
, Y(j)).
(c) Find var(Y(j) − Y(i)).
Let Yİ, Ya, , Yn be independent random variables each having uniform distribu- tion on the interval (0, 6) (a) Find the distribution...
I. Let the random variable y have an uniform distribution with minimum value θ = 0 and maximum value θ2-1 and let the random variable U have the form aY +b, where a and b are both constants and a > 0. (a) Using the transformation method, find the probability density function for the random variable U when a 2 and b-4. What distribution does the random variable U have? (b) Using the transformation method, find the probability density function...
S2-R be a random variable on a probability space (LF, P) with the uniform distribution on [1-1,T+름 . Does there exist a random variable Y : Ω → R For each n E N, let Yn such that Y,,-, Y almost surely as n-> oo?
S2-R be a random variable on a probability space (LF, P) with the uniform distribution on [1-1,T+름 . Does there exist a random variable Y : Ω → R For each n E N, let...
Let X be a random variable with the probability density function f(x)= x^3/4 for an interval 0<x<2 (a) What is the support of X? (b) Letting S be the support of X, pick two numbers a, b e S and compute Pa<x<b). Draw a graph that shows an area under the curve y = f() that is equal to this probability. (c) What is Fx (2)? Draw a good graph of y=Fx (I). (d) What is EX? (e) What is...
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