Problem 2 (9 points) Consider a random variable X with pdf given by: (x) 0.06x +0.05...
Consider a random variable X with pdf given by: F(x) = 0.06x+0.05 x= 0,1.5,2,4,5 Find P(2.5<X<7) __________________ Given E[x] = 3.46, find Var[x]= __________________ SHOW ALL WORK ANSWER ALL PARTS
1. (20 points) Consider a random variable X with PDF and a random variable Y with PDF o)(350 e ys0 Given thatX and Y are independent, find the PDF of Z = X + Y. 1. (20 points) Consider a random variable X with PDF and a random variable Y with PDF o)(350 e ys0 Given thatX and Y are independent, find the PDF of Z = X + Y.
(2 points) Below is the partially complete pmf for a random variable X. Enter the missing probability. x 0.5 3.5 5 5.5 P(X) 0.05 0.1 0.15 Find EX. E(X) =
6. Consider a random variable X with pdf given by 0 elsewhere. (a) What is c? Plot the pdf. (b) Plot the cdf of X. Find P(X 0.5< 0.3).
Consider a random variable X with PDF given by f(x)=1/10 for x = 0, 1, 2,...,9. How do we call such a distribution? Then E(X) = 4.5 and Var(X) = 8.25. Consider a sample mean of 50 observations from this distribution. Find the probability that the sample mean is above 4.
3. Consider a continuous random variable X with pdf given by 0, otherwise This is called the exponential distribution with parameter X. (a) Sketch the pdf and show that this is a true pdf by verifying that it integrates to 1 (b) Find P(X < 1) for λ (c) Find P(X > 1.7) for λ : 1
Problem 3 [5 points) (a) [2 points] Let X be an exponential random variable with parameter 1 =1. find the conditional probability P{X>3|X>1). (b) [3 points] Given unit Gaussian CDF (x). For Gaussian random variable Y - N(u,02), write down its Probability Density Function (PDF) [1 point], and express P{Y>u+30} in terms of (x) [2 points)
Problem 4 (12 points; 2,2|414) Consider a certain machine part. Let X = lifetime (in days) of the machine part Suppose that pdf of the random variable X is given by: 4 xexp -x2 otherwise 4pts The random variable X is said to have a Weibull distribution with parameters " and β ' On average, how many days does the machine part last? E(X)- 4pts 4pts Find the probability the machine part survives over 25 days? P(X> 25)-
Problem 9: Suppose X is a continuous random variable, uniformly distributed between 2 and 14. a. Find P(X <5) b. Find P(3<X<10) c. Find P(X 2 9)
L.9) Hand calculation of expected values a) The pdf of a random variable X with the Exponential[u] distribution is for 0x Calculate by hand the expected value of X b) The pdf of a random variable X with the Beta[2, 3] distribution is 12 ( 1-xf x for。S$ 1. Calculate by hand the expected value of X. c) Go with numbers a and b with a<b. Calculate by hand the expected value and variance of a random variable uniformly distributed...