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Final
91. 01. 16
1. (20%)
Suppose the number of cars that arrive at a car wash is described by a
Poisson distribution with a mean of 10 cars/per hour.
(a) What is the probability that no car arrives within 2 hours?
(b) What is the probability that 5 cars arrive within 20 minutes?
(c) What is the probability that the time between the arrival is less than 30
miniutes?
2. (30%)
Given that Z is a standard normal random variable and X is a normal
random variable with mean 2 and variance 4.
(a)
P(1.2  Z  2.5)
(b)
P(2.2  Z  1.24)
(c)
P(1  Z  c)  0.75 . Find c.
(d)
P(1  X  4) .
(e)
P(c  X )  0.84 . Find c.
3. (20%)
The discrete random variable X with the following probability
distribution function
f ( x) 
cx
, x  1, 2, 3, 4 .
10
(a) Find c.
(b) Compute the mean of X.
(c) Compute the variance of X.
4.(20%)
Consider a sample with the following data:
3
12
4
7
14
6
1
2
9
11
(a) Compute the mean.
(b) Compute the standard deviation.
(c) Compute the median.
(d) Determine the interquartile range.
5. (10%)
A survey of business students indicated that students who had spent at
least 3 hours studying per day had a probability of 0.85 of average
scoring above 80. Student who did not spend at least 3 hours studying per
day had a probability of 0.1 of average scoring above 80. It has been
determined that 5% of the business students had spent at least 3 hours
studying per day. Given that a student average scored above 85, what is
the probability that he/she had spent 3 hours studying per day?
6.
(a) Let X be a continuous random variable with probability density
function
f ( x )  x, 0  x  c
 1, c  x  1.5
 0, otherwise
.
Find c and mean of X.
(b) Let X be a Poisson random variable with the probability distribution
function
e q q i
P( X  i ) 
, i  0,1,2, .
i!
Show that
E( X )  q .
2