Question

A sniper is put to test. His aim on a target is known to have a deviation inthe horizontal direction characterized by the normal distribution N(μ=0, σ^2=25 cm^2.) If a single shot of the sniper is known not to have deviated more than 10 cm to the right,what is the probability that the same shot remains within the 5 cm neighborhood of the target? Express the answer in terms of the Q-function,i.e.,the survival function of astandard normal random variable with N(μ=0, σ^=1)?.
Let us say that in the previous question, we are provided withonly the pieceof information that the variance of the sniper’s horizontaldeviation is σ^2=25 cm^2. Can we compute a bound on the probability that the same horizontal deviation remains within 5 cm in this case? And if yes, what will that bound be, and will it bean upper or lowerbound? Explain.

Answer 1

If we have only variance info :

consider two cases :

As seen in both cases, area shaded will be lower than the area around the mean which is the probability when mean of

deviation is zero, so clearly we will have an upper bound on the probability in this case i.e. probty cannot exceed certain value.