For the 9 problems in the assignment the variables values are: a=2, b=13, c=4, d=5, e=6.

1. If there are [10+d]  students in the class, what is the probability that two or more will have the same last 2 digits in their social security numbers (in the same order)? Assume that all digits 0 to 9 are equally likely, and give your answer to within ± 0.000001. (Hint: look up the "birthday problem" in a basic statistics book or on the Internet.)

For Problems 2 and 3, assume X is a Gaussian random variable with mean a and standard deviation 3b. Give answers accurate to ± 0.001.

2. What is the probability, to within ±0.001, that the value of  X lies between c and 2c?

3. If X is known to be greater than e, what is the probability, to within ±0.001, that the value of X lies between c and 2c?

4. Basket #1 holds (b-3) eggs; basket #2 holds 2 eggs; and basket #3 holds 1 egg. In how many ways, exactly, can b eggs be sorted into the baskets? (For example, if b = 5 and the eggs are labeled E1, E2, E3, E4, E5, then one way of sorting the eggs would be {E1 & E2} in basket #1, {E3 & E4} in basket #2, and {E5} in basket #3.

This is no different from the sorting {E2 & E1} in basket #1, {E3 & E4} in basket #2, and {E5} in basket #3.

But it is different from {E3 & E4} in basket #1, {E1 & E2} in basket #2, and {E5} in basket #3.

For Problems 5-7 assume X(t) is a zero-mean stationary Gaussian process with autocorrelation function R_X(t₁, t₂) = be^-|t_1-t_2|a. Give answers accurate to ± 0.001.

5. What is the probability that the value of X(d) lies between c and d?

6. If X(c) is measured and found to have the value b, what is the probability that the value of X(d) lies between c and d?

7. What is the probability that X(c) and X(d) both lie between 0 and d?

8. The random process {X(n), n = 0, 1, 2, ...} is created by starting with the deterministic repeating sequence

X(0)=a,  X(1)=b,  X(2)=c,  X(3)=a,  X(4)=b,  X(5)=c,  a , b , c , a , b , c ,   ...

and adding ±d² to each term according to the outcome of a coin flip (+d² for heads, -d² for tails). So if the sequence of (independent) coin flips turned out to be HHTTHT..., the random process outcomes would be (a+d²)  (b+d²)  (c-d²)  (a-d²)  (b+d²)  (c-d²)  ....

What are the following auto-correlations:

RX(0, 0) ≡ E{X(0) X(0)}

RX(1, 1) ≡ E{X(1) X(1)}

RX(2, 2) ≡ E{X(2) X(2)}

RX(0, 1) ≡ E{X(0) X(1)}

RX(1, 0) ≡ E{X(1) X(0)}

RX(0, 2)  ≡ E{X(0) X(2)}    ?

9. Look at the first two outcomes in the previous problem, X(0) and X(1). The random variable Z is defined as X(0) - X(1). MINUS, NOT PLUS! What is the standard deviation σ (not the variance) of Z?

Attachment:- Assignment File.rar