## Question

A hip joint replacement part is being stress-tested in a laboratory. The probability of successfully completing the test is 0.90. Eight randomly and independently chosen parts are tested.

1. Let X be the number of parts completing the test successfully. Graph the probability distribution of X and indicate whether the distribution is skewed or symmetric.
2. Find P(4<=X<=5).
3. Find the probability that at most 2 parts failed the test.

Random variable X follows binomial distribution with probability of success p=0.9.

P(X=x)=([8],[x])0.9^x(1-0.9)^(8-x)=([8],[x])0.9^x0.1^(8-x), 0<=x<=8.

1. Random variable X follows binomial distribution:

P(X=0)=([8],[0])0.9^0 0.1^8=10^-8.

P(X=1)=([8],[1])0.9^1 0.1^7~~7.2*10^-7.

P(X=2)=([8],[2])0.9^2 0.1^6~~2.268*10^-5.

P(X=3)=([8],[3])0.9^3 0.1^5~~4.0824*10^-4.

P(X=4)=([8],[4])0.9^4 0.1^4=0.0045927.

P(X=5)=([8],[5])0.9^5 0.1^3=0.03306744.

P(X=6)=([8],[6])0.9^6 0.1^2=0.148800348.

P(X=7)=([8],[7])0.9^7 0.1^1=0.38263752.

P(X=8)=([8],[8])0.9^8 0.1^0=0.43046721.

Probability distribution is shown.

As can be seen distribution is skewed to the left (negative skew).

2. P(4<=X<=5)=P(X=4)+P(X=5)=0.0045927+0.03306744=0.03766014.

3. Probability that at most two parts will fail test equals to probability that at least 8-2=6 parts will pass test succesfully:

P(X>=6)=P(X=6)+P(X=7)+P(X=8)=

=0.14880348+0.38263752+0.43046721=0.96190821.