3. Two coins are simultaneously tossed until one of them comes up a head and the other a tail. The first coin comes up a head with probability p and the second with probability q. All tosses are assumed independent. (a) Find the PMF, the expected value, and the variance of the number of tosses. (b) What is the probability that the last toss of the first coin is a head

Step-by-step explanation:

From above-given information

Consider p to be the probability that the first coin comes up as head

Consider q to be the probability that the second coin comes up as tail.

Provided all tosses seems to be independent.

Then;

Let X represent no. of tosses, where X follows a geometric distribution with parameter;

p' = p(1-q) +q(1-p)

with probability mass function (pmf)

Thus, the mean (expected value) and the standard deviation can be computed as:

(b)

Suppose X represents that the first coin is head;

and Y represent the last toss

Therefore, the required probability is:

f

step-by-step explanation:

f

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3