# Tag Archives: JCM_math230_HW9_F22

## Cognitive Dissonance Among Monkeys

Assume that each monkey has a strong preference between red, green, and blue M&M’s. Further, assume that the possible orderings of the preferences are equally distributed in the population. That is to say that each of the 6 possible orderings  ( R>G>B or R>B>G or B>R>G or B>G>R or G>B>R or G>R>B) are found with equal frequency in the population. Lastly assume that when presented with two M&Ms of different colors they always eat the M&M with the color they prefer.

In an experiment, a random monkey is chosen from the population and presented with a Red and a Green M&M. In the first round, the monkey eats the one based on their personal preference between the colors. The remaining M&M is left on the table and a Blue M&M is added so that there are again two M&M’s on the table. In the second round, the monkey again chooses to eat one of the M&M’s based on their color preference.

1. What is the chance that the red M&M is not eaten in the first round?
2. What is the chance that the green M&M is not eaten in the first round?
3. What is the chance that the Blue M&M is not eaten in the second round?

[Mattingly 2022]

## Closest Point

Consider a Poisson random scatter of points in a plane with mean intensity $$\lambda$$ per unit area. Let $$R$$ be the distance from zero to the closest point of the scatter.

1. Find a formula for the c.d.f. and the density of $$R$$ and sketch their graphs.
2. Show that $$\sqrt{2 \lambda \pi} R$$ has the Rayleigh distribution.
3. Find the mean and mode of $$R$$.

[pitman p 389, # 21]

## An example of min and change of variable

Suppose $$R_1$$ and $$R_2$$ are two independent random variables with the same density function

$f(x)=x\exp(-{\textstyle \frac12 }x^2)$

for $$x\geq 0$$. Find

1. the density of $$Y=\min(R_1,R_2)$$;
2. the density of $$Y^2$$

[Pitman p. 336 #21]

## Min, Max, and Exponential

Let $$X_1$$ and $$X_2$$ be random variables and let $$M=\mathrm{max}(X_1,X_2)$$ and $$N=\mathrm{min}(X_1,X_2)$$.

1. Argue that the event $$\{ M \leq x\}$$ is the same as the event   $$\{X_1 \leq x, X_2 \leq x\}$$ and similarly that t the event $$\{ N > x\}$$ is the same as the event   $$\{X_1 > x, X_2 > x\}$$.
2. Now assume that the $$X_1$$ and $$X_2$$ are independent and distributed with c.d.f. $$F_1(x)$$ and $$F_2(x)$$ respectively . Find the c.d.f. of $$M$$ and the c.d.f. of $$N$$ using the proceeding observation.
3. Now assume that $$X_1$$ and $$X_2$$ are independently and exponentially  distributed with parameters $$\lambda_1$$ and $$\lambda_2$$ respectively. Show that $$N$$ is distributed exponentially and identify the parameter  in the exponential distribution of $$N$$.
4. The route to a certain remote island contains 4 bridges. If the time to collapse of each bridge is exponential distributed with mean 20 years and is independent of the other bridges, what is the distribution of the time until the road is impassable because one of the  bridges has collapsed.

[Jonathan Mattingly]

## Change of Variable: Gaussian

Let $$Z$$  be a standard Normal random variable (ie with distribution $$N(0,1)$$). Find the formula for the density of each of the following random variables.

1. 3Z+5
2. $$|Z|$$
3. $$Z^2$$
4. $$\frac1Z$$
5. $$\frac1{Z^2}$$

[based on Pitman p. 310, #10]

## Change of Variable: Uniform

Find the density of :

1. $$U^2$$ if $$U$$ is uniform(0,1).
2. $$U^2$$ if $$U$$ is uniform(-1,1).
3. $$U^2$$ if $$U$$ is uniform(-2,1).