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QJEEAdv24Ma1c2e: JEE Advanced 2024 Mathematics - Class 12 JEEAdv24Ma1c2e Solution: Let be a random variable, and let denote the probability that…

Let XX be a random variable, and let P(X=x)P(X=x) denote the probability that XX takes the value xx . Suppose that the points (x,P(X=x)),x=0,1,2,3,4(x, P(X=x)), x=0,1,2,3,4 , lie on a fixed straight line in the xyx y -plane, and P(X=x)=0P(X=x)=0 for all x∈R−{0,1,2,3,4}x \in \mathbb{R}-\{0,1,2,3,4\} . If the mean of XX is 52\frac{5}{2} , and the variance of XX is α\alpha , then the value of 24α24 \alpha is _____________.

Step-by-Step Verified Solution

Let equation of line is y=mx+c\mathrm{y=mx+c}

x\mathrm{x} 0 1 2 3 4 R−{0,1,2,3,4}\mathrm{R-\{0,1,2,3,4\}}
P(x)\mathrm{P(x)} c\mathrm{c} m+c\mathrm{m+c} 2m+c\mathrm{2m+c} 3m+c\mathrm{3m+c} 4m+c\mathrm{4m+c} 0

∑x=04(mx+c)=1⇒10m+5c=1⇒2m+c=15.... (i)\sum_{x=0}^4(m x+c)=1 \Rightarrow 10 m+5 c=1 \Rightarrow 2 m+c=\frac{1}{5}\quad \text{.... (i)}

amp; mean =∑xiPi=∑i=04(mxi+c)⋅xi=30m+10c=52amp;∴3m+c=14…(2)amp; from (1) and (2) m=120,c=110amp;ΣPixi2=∑i=04(mxi+c)x12amp;=∑i=04(mxi3+cxi2)⇒100 m+30c( Now putting m and c)amp;⇒ΣPi2=5+3=8amp; Variance =ΣPixi2−(ΣPixi)2=8−(52)2=74amp;∴24α=42\begin{aligned}& \text { mean }=\sum x_i P_i=\sum_{i=0}^4\left(m x_i+c\right) \cdot x_i=30 m+10 c=\frac{5}{2} \\& \therefore 3 m+c=\frac{1}{4} \ldots(2) \\& \text { from (1) and (2) } \mathrm{m}=\frac{1}{20}, \mathrm{c}=\frac{1}{10} \\& \Sigma \mathrm{P}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}^2=\sum_{\mathrm{i}=0}^4\left(\mathrm{mx}_{\mathrm{i}}+\mathrm{c}\right) \mathrm{x}_1^2 \\& =\sum_{\mathrm{i}=0}^4\left(\mathrm{mx}_{\mathrm{i}}^3+c \mathrm{x}_{\mathrm{i}}^2\right) \Rightarrow 100 \mathrm{~m}+30 \mathrm{c}(\text { Now putting } \mathrm{m} \text { and } \mathrm{c}) \\& \Rightarrow \Sigma \mathrm{P}_{\mathrm{i}}^2=5+3=8 \\& \text { Variance }=\Sigma \mathrm{P}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}^2-\left(\Sigma \mathrm{P}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}\right)^2=8-\left(\frac{5}{2}\right)^2=\frac{7}{4} \\& \therefore 24 \alpha=42\end{aligned}

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