Question

discrete random variable X has the following probability distribution: x 13 18 20 24 27 P ( x ) 0.22 0.25 0.20 0.17 0.16 Compute each of the following quantities. P ( 18 ) . P(X > 18). P(X ≤ 18). The mean μ of X. The variance σ 2 of X. The standard deviation σ of X.

Answer 1

Answer:

(a) P(X = 18) = 0.25

(b) P(X > 18) = 0.53

(c) P(X ≤ 18) = 0.47

(d) Mean = 19.76

(e) Variance = 22.2824

(f) Standard deviation = 4.7204

Step-by-step explanation:

We are given that discrete random variable X has the following probability distribution:

            X                    P (x)             X * P(x)            $X^{2}$             $X^{2}$ * P(x)

           13                    0.22              2.86              169              37.18

           18                    0.25              4.5                324               81

           20                   0.20               4                  400               80

           24                    0.17              4.08              576              97.92

           27                    0.16              4.32              729             116.64

(a) P ( X = 18) = P(x) corresponding to X = 18 i.e. 0.25

     Therefore, P(X = 18) = 0.25

(b) P(X > 18) = 1 - P(X = 18) - P(X = 13) = 1 - 0.25 - 0.22 = 0.53

(c) P(X <= 18) = P(X = 13) + P(X = 18) = 0.22 + 0.25 = 0.47

(d) Mean of X, $\mu$ = ∑X * P(x) ÷ ∑P(x) = (2.86 + 4.5 + 4 + 4.08 + 4.32) ÷ 1

                                                         = 19.76

(e) Variance of X, $\sigma^{2}$ = ∑$X^{2}$ * P(x) - $(\sum X * P(x))^{2}$

                                 = 412.74 - $19.76^{2}$ = 22.2824

(f) Standard deviation of X, $\sigma$ = $\sqrt{variance}$ = $\sqrt{22.2824}$ = 4.7204 .