The distribution function of the random variable X isgiven byF(x) = 1 − (1 + x)e−x for x > 00 for x F 0Find P(X F 2),P(1 < X < 3), and P(X > 4).

Q: Let X be a continous random variable with a cumulative distribution function (cdf) as follows 0, r 6…

A: From the expression of Y it can be observed that the value of Y will be zero or less than 0 only if…

Q: A continuous random variable X has probability distribution function f(x) , where f(x) = k(1 − x2)…

A: The k value is calculated as follows:

Q: Let X and Y be independent random variables distributed as N(-1, 1) and N(1, 1), respectively. Find…

A: X follows a  normal distribution with , µ= -1 , σ= 1. Y follows a  normal distribution with , µ= 1 ,…

Q: For a continuous random variable X you are told that P(X > 0) = 1.    Additionally, you are told…

A: Given that for a continuous random variable X,  PX&gt;0=1  and   eXd=d+1002,  for all d&gt;0

Q: (i) the maximum value of p.d.f. (ii) expectation of X, E(X) (iii) variance of X.

A:

Q: A random variable X takes values 0,1,2 where E(x) =1.5, var(x)=0.25 then E (3x-x²+4) =

A: Given that E(X) = 1.5, and var(X) = 0.25 The var(X) is defined as Var(X) = E(X2) - (E(X))2

Q: Let X be a discrete random variable. (i) Prove that V ar(X) = E(X2) ? (EX)2. Hint: Use the…

A: We want to prove both results

Q: Show that the mean x of a sample of size n from the distribution. e-x/0 f (x)= 0<x< 0 elsewhere is…

A: Here given f(x) which is the distribution of exponential distribution.  We know the mean and…

Q: A random variable X has a normal distribution N(4,9). (1) Find V(X) and E(X²). (2) Let + be the cdf…

A: The normal distribution of the random variable is N(4,9).

Q: Suppose X is a uniform random variable on the interval (1, 3), and E(X) = 2. Find the probability…

A: X follows uniform (1, 3). That is, fx=13-1=12VX=b-a212SDX=b-a12=3-112=212=13

Q: A random variable X has only two values a and b with P(X = a) = p , P(X = b) = q (p + q = 1). Find…

A:

Q: 3 1/8 (a) Find the marginal distribution of X1 1/4

A:

Q: X is a random variable with the probability distribution f(X)= :X= 0,1,2. Then the mean of X is…

A: Given function is: f(X)=X2;X=0,1,2

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!