Question

In: Statistics and Probability

Let A be a non-negative random variable (A>0) a) If A is discrete, show that E[A]...

Let A be a non-negative random variable (A>0)

a) If A is discrete, show that E[A] >= 0

b) If A is continous , show E[X]>=0

Expert Solution

a)

A discrete random variable A has a countable number of possible values.

Example: Two dice are thrown. Let A represent the sum of two dice. In this case, A is non negtative.

Then the probability distribution of X is as follows:

A	2	3	4	5	6	7	8	9	10	11	12
P (A)	1/36	2/36	3/36	4/36	5/36	6/36	5/36	4/36	3/36	2/36	1/36

E(A) = ∑ A∙ P(A)

= [ 2∙(1/36) ] + [ 3∙(2/36) ] + [ 4∙(3/36) ] + [ 5∙(4/36) ] + [ 6∙(5/36) ] + [ 7∙(6/36) ] + [ 8∙(5/36) ] +

[ 9∙(4/36) ] + [ 10∙(3/36) ] + [ 11∙(2/36) ] + [ 12∙(1/36) ]

= [ 2 + 6 + 12 + 20 + 30 + 42 + 40 + 36 + 30 + 22 + 12 ] / 36

= 252 / 36

E(A) = 7

b)

Let A = Continuous random variable and it is defined as

A = x²; 0 < x < 3

= 0 ; Otherwise

E[x] =

=

= 1/4 { ( x⁴ )_{3 -} ( x⁴ )₀}

= 1 / 4 [ 3⁴]

E[X] = 81 / 4

E[X] = 20.25

In both the cases Expected value is greater than zero.

orchestra answered 1 year ago

Let X be a discrete random variable that takes value -2, -1, 0, 1, 2 each...

Let X be a discrete random variable that takes value -2, -1, 0, 1, 2 each with probability 1/5. Let Y=X2 a) Find the possible values of Y. Construct a joint probability distribution table for X and Y. Include the marginal probabilities. b) Find E(X) and E(Y). c) Show that X and Y are not independent.

Let T > 0 be a continuous random variable with cumulative hazard function H(·). Show that...

Let T > 0 be a continuous random variable with cumulative hazard function H(·). Show that H(T)∼Exp(1),where Exp(λ) in general denotes the exponential distribution with rate parameter λ. Hints: (a) You can use the following fact without proof: For any continuous random variable T with cumulative distribution function F(·), F(T) ∼ Unif(0,1). Hence S(T) ∼ Unif(0,1). (b) Use the relationship between S(t) and H(t) to derive that Pr(H(T) > x)= e−x.

Let X be a random variable with CDF F(x) = e-e(µ-x)/β, where β > 0 and...

Let X be a random variable with CDF F(x) = e-e(µ-x)/β, where β > 0 and -∞ < µ, x < ∞. 1. What is the median of X? 2. Obtain the PDF of X. Use R to plot, in the range -10<x<30, the pdf for µ = 2, β = 5. 3. Draw a random sample of size 1000 from f(x) for µ = 2, β = 5 and draw a histogram of the values in the random sample...

Let discrete random variable X be the number of flips of a biased coin required to...

Let discrete random variable X be the number of flips of a biased coin required to get tails, where P(tails) = 1/3 . a) Calculate the probability for every value of X from 1 to 10. b) Sketch a plot of the p.m.f. of X for the first 10 flips. c) Sketch a plot the c.d.f. of X for the first 10 flips.

The probability distribution of a discrete random variable x is shown below. X 0 &

The probability distribution of a discrete random variable x is shown below. X 0 1 2 3 P(X) 0.25 0.40 0.20 0.15 What is the standard deviation of x? a. 0.9875 b. 3.0000 c. 0.5623 d. 0.9937 e. 0.6000 Each of the following are characteristics of the sampling distribution of the mean except: a. The standard deviation of the sampling distribution of the mean is referred to as the standard error. b. If the original population is not normally distributed, the sampling distribution of the mean will also...

Suppose that random variable X 0 = (X1, X2) is such that E[X 0 ] =...

Suppose that random variable X 0 = (X1, X2) is such that E[X 0 ] = (µ1, µ2) and var[X] = σ11 σ12 σ12 σ22 . (a matrix) (i) Let Y = a + bX1 + cX2. Obtain an expression for the mean and variance of Y . (ii) Let Y = a + BX where a' = (a1, a2) B = b11 b12 0 b22 (a matrix). Obtain an expression for the mean and variance of Y . (ii)...

1. A coin is tossed 3 times. Let x be the random discrete variable representing the...

1. A coin is tossed 3 times. Let x be the random discrete variable representing the number of times tails comes up. a) Create a sample space for the event; b) Create a probability distribution table for the discrete variable x; c) Calculate the expected value for x. 2. For the data below, representing a sample of times (in minutes) students spend solving a certain Statistics problem, find P35, range, Q2 and IQR. 3.0, 3.2, 4.6, 5.2 3.2, 3.5...

Q) Let Xbe a discrete random variable representing the maximum value of the two numbers on...

Q) Let Xbe a discrete random variable representing the maximum value of the two numbers on throwing two identical balanced dice for one time only. Then: a) Find the possible values of the random variable X for the following cases: b) Determine the probability mass function P (X = ·). c) Draw the graphical representation of the probability mass function P (X = ·). d) Determine the distribution functionF . X e) Sketch the functions in part (a). f) Calculate the...

Determine whether the value is a discrete random variable, continuous random variable, or not a random...

Determine whether the value is a discrete random variable, continuous random variable, or not a random variable. a. The number of bald eagles in a countrynumber of bald eagles in a country b. The number of points scored during a basketball gamenumber of points scored during a basketball game c. The response to the survey question "Did you smoke in the last week question mark "response to the survey question "Did you smoke in the last week?" d. The number...

Let X and Y be two discrete random variables with a common mgf of e^(4t). After...

Let X and Y be two discrete random variables with a common mgf of e^(4t). After some analysis you conclude, X = 4 and Y = 6. Is this a valid claim? Why or why not?