Independent solution

How to solve this Discrete Distributions question

Setup

Setup

Compute the mean and second moment from the discrete probability table, then use them to determine the interval within one standard deviation of the mean.

E[X]=∑xxpxE[X]=\sum_xx p_x
E[X2]=∑xx2pxE[X^2]=\sum_xx^2p_x

Model

Model

The table gives first moment 55 and second raw moment 3500, so the standard deviation is the square root of 3500 minus 55 squared.

E[X]=55E[X]=55
E[X2]=3500E[X^2]=3500
σ=3500−552=21.7945\sigma=\sqrt{3500-55^2}=21.7945

Compute

Compute

The one-standard-deviation interval is [33.2055,76.7945]. Exactly the support values 40, 50, 60, and 70 lie inside it, with total mass 0.45.

55−21.7945≤X≤55+21.794555-21.7945\le X\le55+21.7945
X∈{40,50,60,70}X\in\{40,50,60,70\}
0.05+0.20+0.10+0.10=0.450.05+0.20+0.10+0.10=0.45

Answer

Answer

Therefore 45% of the distribution lies within one standard deviation, corresponding to choice A.

45%(A)\boxed{45\%\quad\text{(A)}}