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Midterm 2 TYPE II: MA 3210, Spring 2000

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.

$\frac{1}{3}$ of cars are equipped with airbags. Let X be the number of the cars sold that are equipped with airbags among the randomly selected 4 cars.

(a)

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

(b)

Find the expected number of cars equipped with airbags.

(c)

Find the cumulative density function F(x) of X.

(d)

Find the probabilty that at least 2 of them are equipped with airbags.

2.

A comittee of 2 is to be selected from 2 chemists and 2 biologists. Let X be the number of biologists in the committee.

(a)

Find the probability density function of X.

(b)

Find the expected value of X.

(c)

Find the variance of X.

3.

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

$$f(x) = \left( \begin{array}{ll} 0 & \mbox{elsewhere} \\ \frac{x}{8} & 0< x < 4 \end{array}$$

(a)

Find P( X >2).

(b)

Find P(0.5 < X < 1).

(c)

Find the cumualtive density function.

(d)

Find the expected value of X.

(e)

Find the variance of X.

(f)

Find the expected value the random variable g(X) = x² + 1.

4.

Let X denote the number of times that a student is sick and Y be the number of the visist to the doctor. Their joint probability is given as

f(x,y)	x	1	2	3
	0	0.1	0.1	0.2
y	1	0	0. 1	0.4
	2	0	0.1	0

(a)

Find the mariginal distribution of X alone.

(b)

Find the mariginal distribution of Y alone.

(c)

Given that he made a visit to the doctor twice, find the probability that he was sick once.

(d)

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

(e)

Find the covariance $\sigma_{XY}$ of X and Y.

5.

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

$$f(x,y) = \left( \begin{array}{ll} 0 & \mbox{elsewhere} \\ \frac{ 6- x -y}{8}, & 0< x < 2, 2 < y < 4 \end{array}$$

(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.

6.

Use the Binomial Distribution table. A coin is biased so that head is three times likely to occur as a tail. If a coin is tossed 15 times,

(a)

what is the probability of getting at least 8 Heads ?

(b)

What is the probabiltiy of having at least 8 Heads but less than or equal to 11 Heads?

(c)

What is the mean and the variance in this model. Here you may use the formula.

7.

A random variable X has a mean $\mu = 100$ and the variance $\sigma^2 = 25$. Using Chebyshev's theorem, find

(a)

P( 85 < X < 115).

(b)

$P(\vert X - 100\vert \ge 20)$.

8.

(a)

Make one probability model which has a binomial distribution.

(b)

Give one example whose probability model is likely to have a positive covariance.

(c)

Give an example of joint probability model whose covairance is likely to be negative.

(d)

Make a problem whith a multinomial distribution.

9.

(Extra Credit) The joint probability density function of random variables X, Y are given as

$$f(x,y) = \left( \begin{array}{ll} 0 & \mbox{elsewhere }\\ e^{-(x + y)}, & x>0, y>0 \end{array}$$

(a)

Find the marginal distribution in Y alone.

(b)

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

(c)

Find the P(X < 1 | Y=1)

(d)

Test if X and Y are statistically independent.