# Expression for the probability density function of the binomial distribution

### Solve the second question of this problem

From the question, we can see that X obeys the binomial distribution B(4, 0.5) and Y obeys the exponential distribution with parameter 2.

The requirement P{4X-Y>10} can first be obtained by analyzing the distributions of X and Y to obtain the distribution of 4X-Y.

For the binomial distribution B(n,p), its expectation and variance are np and np(1-p), respectively. Thus, for X obeying B(4, 0.5), we have:

E(X)=np=4×0.5=2

Var(X)=np(1-p)=2

Since Y obeys an exponential distribution with a parameter of 2, its probability density function is:

f(y)=1/2*exp(-y/2), y>0

Thus, the probability density function of 4X-Y can be obtained as:

f(x,y)=f(x)f(y)=[C(4,x)0.5^4(1-0.5)^(4-x)][1/2exp(-y/2)]

=C(4,x)exp(-y/2)/16

Requiring P{4X-Y&>0 gt;10}, you can take the exponents of both sides of the inequality to get:

exp(4X-Y)>exp(10)

i.e.

4Xexp(-Y/4)>e^(-10)

So we have:

P{4X-Y>10}=P{4Xexp(-Y /4)>e^(-10)}=P(X>10/e(-10)*exp(Y/4))=0.5-0.5*(1-(1-e-(5/2))^4)

The last of these equations uses the fact that X is binomially distributed, or it can be solved directly with the probability mass function of the binomial distribution, as seen below:

P{4X- Y>10}=∑(x=0)^4∫(10+4x)/e^(-10)1/2exp(-y/4)dy

=∑(x=0)^4[1-exp(-(10+4x)/2)]

=[1-exp(-5)]^4

≈0.621

Thus, P{4X -Y>10}≈0.621.