Final Exam, MA 3210, Introduction to Probability and Statistics, spring 2000
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Useful Formula
$\mu=E[x]=\left( \begin{array}{ll} \sum x f(x) & \mbox{ if X is discrete} \\ \int x f(x) dx & \mbox{if X is continuous} \end{array}$
$\sigma^2 = E[(X-\mu)^2] = E(X^2) - E(X)^2=E[X^2] -\mu^2$
$\sigma_{XY}=E[(X-\mu_x)(Y-\mu_Y)]= E[XY] - \mu_X \mu_Y, ~\mu_X=E[X],~\mu_Y=E[Y]$

1.

(a)

P(A|B)=0.5, P(B)=0.7. Find $P(A \cap B)$.

(b)

For a random variable $X, ~~P(X \le 10) = 0.4$. Find P(X >10).

(c)

What does it mean that two events A, B are independent?

(d)

How many permutations are possible to arrange the letters in "peer"?

2.

The probability that OW students like a dog is 50% and the probability that OW students like a cat is $40 \%$, and the probability that OW students like both a dog and a cat $25 \%$.

(a)

Find the probability that a randomly chosen OW student like a cat or a dog.

(b)

Find the probability that a randomly chosen OW student like both cat and dog.

3.

In a poker hand consisting of 5 cards, find the probability of holding

(a)

3 aces and 2 King.

(b)

2 spades and 2 club and one heart.

4.

Judy tosses a coin which has $P(H)= \frac{2}{3}$. If it is Heads, she draws a ball from Bag 1. If it is Tails, she draws a ball from bag 2. Bag I contains 2 red balls and 1 blue balls, Bag II contains 1 red balls and 2 blue balls. R denotes the event of drawing a red ball, B denotes the event of drawing a blue ball and H and T denote the event of getting Head and Tail respectively.

(a)

Complete the probability tree for this experiment.

(b)

Find the probability P(R|H).

(c)

$P(R \cap H)$

(d)

Find P(R).

(e)

She drew a red ball. Find the probability that she got Heads.

5.

Long Island Railroad has 2 lines. L₁, L₂ which make up 60%, and 40% of LIRR service. The data shows that 3 %, 2% of each line are running late.

(a)

If a train is randomly selected, what is the probability that it is running late?

(b)

A train is randomly selected from the ones which run late. What is the probability that it is on line L₂?

6.

It is known that $\frac{2}{3}$ of the cars pass the inspection. Let X be the number of the cars passing inspection. Among the randomly selected 3 cars,

(a)

Find the probability density function f(x) of X.

(b)

Find the probability that at least 2 of them pass the inspection.

(c)

Find the expected number of cars which will pass the inspection.

7.

The probability density function f(x)of a random variable X is given as

$$f(x) = \left( \begin{array}{ll} 0 & \mbox{elsewhere} \\ \frac{x}{2} & 0< x < 4 \end{array}$$

(a)

Find P(-1< X < 1).

(b)

Find the expected value of X.

8.

Let X denote the number of emergency patients admitted to the hospital : 1,2, or 3 times on any given day. Let Y denote the number of times a doctor was called on an emergency call.

f(x,y)	x	1	2	3
	y
	1	0.4	0. 25	0.15
	2	0	0.05	0.15

(a)

Find the probability that $P( X \le 2, Y \le 2 )$.

(b)

Given that a doctor was called on once, find the probability that two patients were admitted.

(c)

Verify that f(x,y) is a joint probability distribution function.

9.

Given the joint density function of a random variable X and Y,

$$f(x,y) = \left( \begin{array}{ll} 0 & \mbox{elsewhere} \\ x + y, & 0< x <1, 0 < y <1 \end{array}$$

(a)

Find P( 1<X<2, Y>3).

(b)

Find the marginal distribution of X, Y.

(c)

Test if X and Y are statistically independent.

10.

A random variable X has a mean $\mu = 20$ and the variance $\sigma^2 = 4$. Using Chebyshev's theorem, find P( 14< X < 26).

11.

Write down the probability distribution density function of the following.

(a)

A Poisson distribution whose expected number of arrivals per unit time is 5.

(b)

discrete uniform distribution over 5 numbers $\{1,2,3,4,5\}$

(c)

Normal distribution with $\mu = 15, ~\sigma = 2$.

(d)

Standard Normal Distribution.

12.

In this problem, use the statistical tables. In each problem, indicate which distribution table you are using (Binomial, Poisson, or Normal).

(a)

A coin is biased so that P(H)=0.4. Tom tosses this coin 10 times. What is the probability of having 4 or more Heads but less than or equal to 6 Heads?

(b)

Given a normal distribution with $\mu=30$, $\sigma=3$, Find P(21<X<27).

(c)

It is known that one out of 100 patients is allergic to grass. What is the probability that a randomly chosen 1000 patients will yield fewer than 9 patients who are allergic to grass?

(d)

A process yields 20 % defective items. If 100 items are randomly selected from the process, what is the probability that the number of defective items exceeds 16 but less than 24?