# Binomial random variable and binomial distribution

Suppose that x is a binomial random variable with n = 5, p = 0.2, and q = 0.8.

(a) For each value of x, calculate p(x), and graph the binomial distribution. (Round final answers to 5 decimal places.)

p(0) = , p(1) = , p(2) = , p (3) = ,

p(4) = , p(5) =

(b) Find P(x = 3). (Round final answer to 5 decimal places.)

p(x=3)

(c) Find P(x ≤ 3). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x ≤ 3)

(d) Find P(x < 3). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x < 3) = P(x ≤ 2)

(e) Find P(x ≥ 4). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x ≥ 4)

(f) Find P(x > 2). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x > 2)

(g) Use the probabilities you computed in part b to calculate the mean, μx, the variance, σ 2x, and the standard deviation, σx, of this binomial distribution. Show that the formulas for μx , σ 2x, and σx given in this section give the same results. (Do not round intermediate calculations. Round final answers to µx and σ 2x in to 2 decimal places, and σx in to 6 decimal places.)

µx

σ2x

σx

(h) Calculate the interval [μx ± 2σx]. Use the probabilities of part b to find the probability that x will be in this interval. Hint: When calculating probability, round up the lower interval to next whole number and round down the upper interval to previous whole number. (Round your answers to 5 decimal places. A negative sign should be used instead of parentheses.)

The interval is [, ].

P( ≤ x ≤ ) =

#### Solution Preview

Suppose that x is a binomial random variable with n = 5, p = 0.2, and q = 0.8.

(a) For each value of x, calculate p(x), and graph the binomial distribution. (Round final answers to 5 decimal places.)

p(0) =5C0*0.2^0*(1-0.2)^5=0.8^5=0.32768

P(1)=5C1*0.2^1*(1-0.2)^(5-1)=5*0.2*0.8^4=0.4096

P(2)=5C2*0.2^2*0.8^3=5*4/2*0.04*0.8^3=0.2048

P(3)=5C3*0.2^3*0.8^2=5*4/2*0.008*0.8^2=0.0512

P(4)=5C4*0.2^4*0.8^1=5*0.2^4*0.8=0.0064

P(5)=5C5*0.2^5*0.8^0=0.00032

(b) Find P(x = 3). (Round final answer to 5 decimal places.)

P(x=3)=0.0512

(c) Find P(x ≤ 3). (Do ...

#### Solution Summary

The expert examines binomial random variables and binomial distirbutions.