Let X1 and X2 be IID exponential with parameter  > 0. Determine the distribution ofY = X1=(X1 + X2).

A person arrives at the side of a single lane, one way street as a car passes by. Consider this time as time 0. There is no traffic lights and no pedestrian lanes in the nearby area. He stands there watching cars pass by. The number of cars to pass by within a period of time cane be modeled using a poison distribution. Suppose the cars pass by the person at a rate of 4.5 cars per minute. Assume that the cars pass by instantaneously. what is the probability that no cars pass by him within a 2 minute window b) if the person had to guess how many cars would pass by in the next five minutes, what number should he guess? Why? c) what is the distribution for the time until the first car passes by him(time is continuous). What is the expected time until the first car passes by him?

A model for normal human body temperature, X, when measured orally in °F, is that it is normally distributed, X ~ N (98.2,0.5184). (i) According to the model, what proportion of people have a normal body temperature of 99 °F or more? (ii) Find the normal body temperature such that, according to the model, only 10% of people have a lower normal body temperature. (iii) Let W denote normal human body temperature, when measured orally in °C. Given that W = (X - 32) and X ~ N(98.2,0.5184), what is the distribution of W? (iv) According to the model that you just derived for W, what proportion of people have a normal body temperature of between 36 °C and 36.8 °C?

Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.

Suppose that X follows a lognormal distribution with w^2=4 and E(X)=10,000 a. Find the parameter theta for the distribution. Use 4 digits after the decimal point: b. Find the standard deviation of the process: Use 4 digits after the decimal point:

Let X ~ Uniform(a, b). Find Expected value EX.

Given pdf f (x) = 1.5x2  for −1< x < 1. Determine variance of X.

The type of battery in Jim's laptop has a lifetime X (in years) which follows a Weibull distribution with parameters a = 2 and B = 4. The type of battery in Jim's tablet has a lifetime Y (in years) which follows an exponential distribution with parameter ) =1/4. Find E(XY 2Y). (Answer as a decimal number, and round to 2 decimal places).