Let px (x) be the pmf of a random variable X. Find the cdf F(z) of X and its graph along with that of px(x) if:

Q: Suppose X is a random variable, whose pdf is defined as follows: 2x = (²x) (u(x) - u(x − 3)) where…

A: The solution is given below.Explanation:Given:Given the probability density function (pdf) for a…

Q: If fx₁x₂ (x₁, x₂) is a joint density function, then find the joint density function of Y, and Y₂

A:

Q: Let u(X) be a nonnegative function of a random variable X. It can be shown that  P(u(X) >= c) = a)…

A: It is an important part of statistics. It is widely used.

Q: Consider a Brownian motion {X(t), t≥ 0} with parameter o = 2. Find the E[X(t)]] (the expectation of…

A: Solution: From the given information, {X (t),t≥0} is a Brownian motion with a parameter σ = 2.

Q: manufacturing company employs two devices to inspect output for quality control purposes. The first…

A:

Q: If E and F are two independent events, such that P (E ∩ F ) = 1/6 , P (E' ∩ F' ) = 1/3 and (P (E) −…

A:

Q: Let the random variable X have the moment generating function e³t M(t) = 1-t²-1<t<1 What are the…

A:

Q: Three components are randomly sampled, one at a time, from a large lot. As each component is…

A: The objective of this question is to find the probability of two specific sequences of successes and…

Q: Suppose X1, X2, and X3 are independent Bernoulli random variables with success prob- abilities p1,…

A: E(X1E(X2)=p2E(X3)=p3

Q: Consider the sample space S= {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)}, where each point is assumed…

A: Given sample space is, S=(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4). Each point is equally likely.…

Q: Let fix) = 2x, 0 <x<1, zero elsewhere, be the pdf of X. (a) Compute E(X) and Yar(X) (b) Find mean…

A: Expected Value and variance: If X is given as the random variable with the probability density…

Q: The CDF F(x) of a random variable X is given by; F(x) = { 0, if x 3 } i) the mean and variance of…

A: Here,fx=dfxdx=dcx-14dx=4cx-13                                    ;1&lt;x≤3   Now, find c using the…

Q: A) Suppose that X is a random variable with if 0 < x < 2 fx(x) = otherwise

A: We have to find the var(x)  Using formula  Var(x) = E(x^2) - [E(x)]^2

Q: Suppose that X₁, X2, X3 are independent and identically distributed random variables with…

A: Solution

Q: Suppose that X, and X2 are aiscrete random variables with joint pdf of the form f(x1,x2) = c(x1 +…

A:

Q: The diameter of a round rock in a bucket, can be messured in mm and considered a random variable X…

A: We want find  Mean= E(x) , variance = var(x)

Q: Let x be a continuous random variable with PDF 3 x > 1 x4 fx (x) otherwise Find the mean and…

A: Given PDF is fX(x) = 3x4,x ≥ 10,otherwise

Q: A Computer manufacturer wants to estimate the mean number of defective microchips it produces in a…

A: Margin of error is used to obtain the error occurred in the population estimate interval. It depends…

Q: For a random variable x, a new random variable y can be created by applying a linear transformation…

A: We have to find the new mean salary for the employeesfrom the given linear transformation formula…

Q: B) Let the random variable X have the moment generating function e3t M(t) 1- t? for -1<t < 1, What…

A: Solution

Q: (4) If a point x is taken at random on the interval [0, 2a] and all the points are equally likely to…

A: Given :x2a-x&gt;a22

Q: The CDF F(x) of a random variable X is given by; F(x) = { 0, if x 3 } (i)find PDF f(x) (ii) P(|X| <…

A: (i) fx=dFxdx                                  for 1&lt;x≤3=dcx-14dx=cdx-14dx=4cx-14-1=4cx-13…

Q: (4) If a point x is taken at random on the interval [0, 2a] and all the points are equally likely to…

A: the selection of a point in the closed interval [0,2a] is a uniform random variable X.

Q: Let Y₁, Y2,..., Yn denote a random sample of size n from a population with a uniform distribution on…

A:

Q: Let X1, X2, ….,n be iid random variables with common CDF F. Generate the random variable defined by…

A: This method uses the property that the minimum of n i.i.d.  random variables with common CDF F…

Q: Let X be a random variable with pdf: f(x)= k(10 - 2x) for x being [0,5]  a.) Find k b.) Find E(X)…

A: GIVEN DATA : X is the random variable with probability density function fx=k10-2x in the interval of…

Q: benefit, issued to (x). Let Z2 denote the present value of a whole of life n-y term insurance…

A: Given that Z1and Z2 denotes the present value of n-year term insurance benefit and a whole of life…

Q: Let X1, X2, X3 and X4 be exponential(1) random variables. Find the joint distribution of…

A: To find the joint distribution of the random variables , , and, we can use the Jacobian method.…

Q: random process of x(t) is defined by x(t)=Acos(at) + Bsin(at), where A and B are two andom variables…

A:

Q: Show that their covariance is Cry=a o where oy is the variance of X

A:

Q: Let X be a random variable with pdf given by fX(x) = 1/[π(1 + x2)] for all real number x. Prove that…

A: Solution

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!