Let  X  have  the  density 𝑓(𝑥)=2𝑥  if 0≤𝑥 ≤1  and 𝑓(𝑥)=0

otherwise. Show that X has the mean  and the variance 

Find the mean and the variance of the random variable 𝑌 =−2𝑋+3. 

Let the random variable X have the density 𝑓(𝑥)=𝑘 𝑥 𝑖𝑓 0≤𝑥 ≤3.

Find 𝑘. Find  and  such that 𝑃(𝑋 ≤)=0.1 and 𝑃(𝑋 ≤)=0.95.

Find 𝑃(|𝑋−1.8|<0.6). 

Let  X~Uniform(1,  4).  Write  down  its  probability  density  function.

What is the probability that X taks a value between 0 and 2.5. 

Let X is a continuous random variable with probability density function given by

Verify that 𝑓(𝑥) is a probability density function.

Let X is a continuous random variable with probability density function given by 

 Find 𝑃(1≤𝑥 ≤2)   𝑎𝑛𝑑  𝑃(𝑥 >2.

The commuter trains on the Red Line for a city have a waiting time

during peak rush hour periods of ten minutes.

Find the probability of waiting more than 6 minutes.  

Suppose a random variable X is best described by a uniform probability 

distribution  with  range 2 to 6.  Find  the  value  of a that  makes 

the following probability statements true. 

(a) P(x ≤ a) = 0.9. 

(b) P(x < a) = 0.54. 

(c) P(x ≥ a) = 0.01. 

(d) P(x > a) = 0.54. 

(e) P(2 ≤ x ≤ a) = 0.42. 

A small petrol station is supplied with petrol once a week. Assume that its

volume  X  of  potential  sales  (in  units  of  10,  000  liters)

has  the probability density function 𝑓(𝑥)= 6(x − 2)(3 − 𝑥) for 2≤𝑥 ≤3 and 𝑓(𝑥)=  0  otherwise.

Determine  the  mean  and  the  variance  of  this distribution. What capacity

must the tank have for the probability that the tank will be emptied in a given week to be 5%? 

Find the probability that none of the three bulbs in a set of traffic

lights will have to be replaced during the first 1200 hours of operation if

the lifetime X of a bulb (in thousands of hours) is a random variable with probability density

function 𝑓(𝑥)=6[0.25−] when  1≤𝑥 ≤2 and 𝑓(𝑥)=0 otherwise.

You  should  assume  that  the lifetimes of different bulbs are independent.

Historical  evidence  indicates  that  times  between  fatal  accidents  on scheduled

American domestic passenger flights have an approximately exponential distribution.

Assume that the mean time between accidents is 44 days. a) If one of the accidents occurred on July 1 of a randomly

selected year in the study period, what is the probability that another accident occurred that same month? (P(X≤31) b.

What is the variance of the times between accidents?

Suppose X is N(10, 1).

Find

(i) P(X > 10.5),

(ii) P(9.5 < X < 11),  

(iii) 𝑥 such that P(X < 𝑥) = 0.95.

You will need to use Standard Normal tables.

Suppose X is N(−1, 4).

Find 

(a) P(X < 0);

(b) P(X > 1); 

(c) P(−2 < X < 3);   

(d) P(|X + 1| < 1). 

The height of a randomly selected man from a population is normal

with μ = 178cm and 𝜎 = 8cm. What proportion of men from this population are over 185cm tall?

Let X be a continuous random variable with values in [ 0 , 2] and

density . Find the moment generating function  for X if 

Let X be a continuous random variable with values in [ 0 , 1],

uniform density function =1 and moment generating function  

Find  in  terms  of the  moment  generating function for

(a) −𝑋. 

(b) 1 + X. 

(c) 3X. 

(d) a X + b.